Digital Control, Sampling, PLC/DCS and Alarm Management

Scope: sampled measurements; numerical integration; quantization/filtering; alarm management. Cross-cutting case: PLC-based temperature and level control with a digital historian.

What this unit builds

Closed-loop behavior depends on the entire measurement-to-actuation path, rather than on the process alone. A regulator can be mathematically stable and still have unacceptable control quality because the actuator moves too often, the sensor is biased, the sample interval is irregular or the achievable process output is constrained. Performance reporting therefore includes both controlled-variable error and the physical effort used to obtain it.

Lesson sequence

1. [Sampling, aliasing, and execution timing](../lessons/unit-07-lesson-01.html) — A sample interval fixes the fastest frequency that can be distinguished in ideal band-limited sampling. Real plants need anti-alias filtering, timing jitter analysis, and enough samples relative to closed-loop dynamics. Governing relation: `f_s>2 f_max`.

2. [Discrete integration and controller updates](../lessons/unit-07-lesson-02.html) — Digital PI algorithms multiply error by the actual elapsed time. Forward and backward Euler and trapezoidal updates have different numerical characteristics; an implementation should state which it uses. Governing relation: `I[k]=I[k-1]+Ki Ts e[k]`.

3. [Filtering, quantization, and data quality](../lessons/unit-07-lesson-03.html) — A digital exponential filter moderates noise but creates latency, while sensor or ADC quantization causes staircase measurements. Missing or bad-quality samples must be handled by policy, not silently treated as valid zeros. Governing relation: `y_f[k]=alpha y_f[k-1]+(1-alpha)y[k]`.

4. [Alarm hysteresis, bumpless transfer, and interlocks](../lessons/unit-07-lesson-04.html) — Alarms inform operators, while protective interlocks act under defined conditions and should be engineered independently of ordinary regulation. Hysteresis and delay can prevent chattering but cannot replace hazard review. Governing relation: `alarm_on if y>=H; alarm_off if y<=L`.

Model connections and physical interpretation

Sampling, aliasing, and execution timing

A sample interval fixes the fastest frequency that can be distinguished in ideal band-limited sampling. Real plants need anti-alias filtering, timing jitter analysis, and enough samples relative to closed-loop dynamics. Nyquist is a theoretical limit for perfect bandlimited signals; control design typically needs a much faster sample rate than the bare factor of two. The calculation that anchors this lesson is deliberately synthetic: A 0.8 Hz sinusoid sampled at 1 Hz appears as a 0.2 Hz alias in uniformly sampled data; sampling has erased the original frequency information. The reasoning should be checked in the next topic, because a dynamic approximation useful for one type of experiment may be inappropriate when the controlled input, disturbance or measurement location changes.

Discrete integration and controller updates

Digital PI algorithms multiply error by the actual elapsed time. Forward and backward Euler and trapezoidal updates have different numerical characteristics; an implementation should state which it uses. If cycle execution is delayed, pretending Ts remained fixed introduces a hidden integration error. The calculation that anchors this lesson is deliberately synthetic: With Ki=0.1 s⁻¹, Ts=2 s and e=3 maintained for four samples, the accumulated integral increment equals 0.123*4=2.4 output units. The reasoning should be checked in the next topic, because a dynamic approximation useful for one type of experiment may be inappropriate when the controlled input, disturbance or measurement location changes.

Filtering, quantization, and data quality

A digital exponential filter moderates noise but creates latency, while sensor or ADC quantization causes staircase measurements. Missing or bad-quality samples must be handled by policy, not silently treated as valid zeros. Filter alpha is tied to sampling interval; a fixed alpha with changed Ts changes its real time constant. The calculation that anchors this lesson is deliberately synthetic: With alpha=0.8, previous filtered value 20 and new sample 30, updated output is 0.820+0.230=22. A one-count resolution on 0–100 over 10 bits is roughly 0.0977 units. The reasoning should be checked in the next topic, because a dynamic approximation useful for one type of experiment may be inappropriate when the controlled input, disturbance or measurement location changes.

Alarm hysteresis, bumpless transfer, and interlocks

Alarms inform operators, while protective interlocks act under defined conditions and should be engineered independently of ordinary regulation. Hysteresis and delay can prevent chattering but cannot replace hazard review. Thresholds are educational examples, not recommended setpoints for any process; acceptable thresholds require site-specific risk analysis. The calculation that anchors this lesson is deliberately synthetic: An illustrative high-temperature alarm raises at 90 °C and clears at 87 °C. A trace 89→91→89→88→86 °C triggers once and clears only at 86 °C. The reasoning should be checked in the next topic, because a dynamic approximation useful for one type of experiment may be inappropriate when the controlled input, disturbance or measurement location changes.

Unit synthesis: a reproducible experiment

Start from PLC-based temperature and level control with a digital historian. Draw the boundary and place every relevant measurement on it. Give the operating point, consistent units and any assumptions about constant density, heat capacity or sensor behavior. Construct a hypothesis that could be tested with a small safe, synthetic input perturbation. Predict the output sign and a first approximation for the relevant time scale before computing anything.

Next compare a nominal model with one deliberately misspecified version. Change a gain, a lag, a delay or a sensor bias rather than modifying all uncertain features at once. Quantify the difference in output peak, settling time, residual steady error and manipulated-variable excursion. Record when the model ceases to be a good representation of the exercise. A successful classroom exercise is a transparent chain of assumptions and checks, not merely a curve.

Finally design a holdout test. In a process setting this might be a second flow rate, a different operating temperature or a load disturbance that was not included in model fitting. Compare model and measurement on the same time grid, keep a reproducible data table, and explain why any fitted coefficient cannot be assumed constant outside its observed range. Treat alarm and protective requirements as a separate design decision.

Unit knowledge check

1. For Sampling, aliasing, and execution timing, state the terms of f_s>2 f_max, choose a limiting or equilibrium case, and explain when it would be invalid.
2. For Discrete integration and controller updates, state the terms of I[k]=I[k-1]+Ki Ts e[k], choose a limiting or equilibrium case, and explain when it would be invalid.
3. For Filtering, quantization, and data quality, state the terms of y_f[k]=alpha y_f[k-1]+(1-alpha)y[k], choose a limiting or equilibrium case, and explain when it would be invalid.
4. For Alarm hysteresis, bumpless transfer, and interlocks, state the terms of alarm_on if y>=H; alarm_off if y<=L, choose a limiting or equilibrium case, and explain when it would be invalid.

Continue to the [Unit 7 assessment](../assessments/unit-07.html) or [Unit 7 project](../projects/unit-07.html).

Unit 7, Lesson 1: Sampling, aliasing, and execution timing

Unit: Digital Controllers, Data, and Alarm Logic. Physical setting: PLC-based temperature and level control with a digital historian. Reference model: f_s>2 f_max.

Learning outcomes

By the end of this lesson you should be able to (i) explain sampling, aliasing, and execution timing in physical terms, (ii) identify the variables and assumptions of f_s>2 f_max, (iii) reproduce and challenge a numerical benchmark, (iv) recognize how measurement, delay and actuator limits modify the model, and (v) design a specific experiment that can distinguish the current model from an alternative interpretation.

Physical reasoning, not just a formula

A sample interval fixes the fastest frequency that can be distinguished in ideal band-limited sampling. Real plants need anti-alias filtering, timing jitter analysis, and enough samples relative to closed-loop dynamics.

Consider the unit-wide equipment case: PLC-based temperature and level control with a digital historian. In this setting the same numerical trend may have more than one physical cause. A change observed at the analyzer may reflect a real process response, a changed flow residence time or the instrument dynamics. The analyst must first specify which quantity is manipulated, which is observed and which variables are treated as external disturbances. This is particularly important for sampling, aliasing, and execution timing, where confusing a state with a measured output can give a mathematically solvable but physically incorrect model.

The chosen governing expression is f_s>2 f_max. It is a compact model, not a complete plant specification. Terms on opposite sides of an equality must have compatible units and consistent time bases. When a parameter is estimated from one transient, its result belongs to that operating range unless new data demonstrate transferability. The task is to explain the meaning of the relation before substituting numbers and then to compare its predictions with a checkable boundary condition or steady state.

Working the numerical reference case

Synthetic worked example. A 0.8 Hz sinusoid sampled at 1 Hz appears as a 0.2 Hz alias in uniformly sampled data; sampling has erased the original frequency information.

First transcribe every numerical input into a short table with its physical unit. Second, compute the baseline result without a software solver. Third, identify which output should change and which should remain unchanged when the manipulated input is zero. In this sampling, aliasing, and execution timing exercise, the reported figures are not fitted plant data: they are an internally stated teaching benchmark. Any extension to different equipment requires new coefficients and new validation.

A useful sensitivity check is to change the selected coefficient or input by +10% while retaining all other conditions. This is not equivalent to assuming the output must also rise by 10%; exponential responses, ratios, saturation and feedback can change the mapping. Write the original expression, substitute the perturbed value and compare the sign of the result against physical intuition. Next repeat with a −10% change. An asymmetric response is evidence of nonlinearity in the selected mapping, not necessarily a numerical error.

Derivation and model restrictions

Nyquist is a theoretical limit for perfect bandlimited signals; control design typically needs a much faster sample rate than the bare factor of two. To turn a physical inventory into a differential state equation, write accumulation as incoming rate minus outgoing rate plus net production, if relevant. To turn an instrument or actuator into a dynamical element, specify its output, time constant and initial condition. To build a transfer function, take deviations about a steady operating point and verify that the assumed initial deviations are zero. Do not apply frequency-domain algebra blindly to a strongly nonlinear or switching regime.

For sampling, aliasing, and execution timing, the limiting checks should include a zero-change input, a long-time equilibrium where one exists and at least one infeasible input that the physical equipment cannot realize. If a parameter has dimensions of reciprocal time, use it only with a time expressed in matching units. If a parameter multiplies a temperature difference, its unit must convert the difference into a heat rate before the quantity is added to the energy balance. These simple checks detect mistakes that a graph renderer will not.

How to identify the model experimentally

Use the previously studied topic Sampling, aliasing, and execution timing as a source of prior knowledge and the next topic Discrete integration and controller updates as an independent challenge. Define the reference operating point for sampling, aliasing, and execution timing in the process case and record a small safe input sequence, the actual actuator trajectory, the measurement times and instrument range. A step, a pulse and a slow ramp probe different properties, so do not collapse them into one claimed validation. If the fitted output differs from the recorded trace, inspect the timestamp alignment and sensor dynamics before increasing model order.

Where possible, obtain at least two input amplitudes. Normalize the measured output by the change in input and test whether the curves approximately collapse. Failure to collapse can indicate nonlinearity, saturation, temperature-dependent coefficients or changed initial conditions. A second holdout operating point is especially important when the main formula was derived by linearization. Report the residual error and uncertainty rather than relabeling an imperfect approximation as a universal physical law.

Uncertainty, troubleshooting, and design decisions

Failure mode to investigate: An operator trend that aliases oscillation can falsely suggest the equipment is changing slowly. The first diagnostic question is which conservation law, measurement chain or actuator path the proposed explanation implicates. Reconstruct the actual manipulated-variable history and compare it with what the controller requested; a large difference is a plant-side constraint, not a parameter-estimation error that can be fixed by fitting a more elaborate process transfer function.

Decision from the evidence: Compare the same physical signal at 1, 5 and 20 samples per second and calculate apparent frequency. State what the experiment supports and what remains unidentifiable. The decision should include whether the next useful action is to add a sensor, run a second input condition, change an assumed model structure or restrict the admissible operating range. When model outputs could guide actual equipment, independent engineering review and safety analysis are necessary before any implementation.

A challenge to the simplest explanation

Compare the present explanation with Filtering, quantization, and data quality, which appears elsewhere in this unit. Imagine obtaining a smooth measured response that both models could plausibly produce. Choose one additional measurement that would change differently under the two explanations. Define the expected sign and time scale under each. For sampling, aliasing, and execution timing, record not just peak error but also the total actuator travel, disturbances, measurement noise and how the reference signal was generated. This exercise forces you to separate apparent goodness of fit from the ability to predict new conditions.

Numerical implementation and independent checks

The companion [interactive model, Laboratory 14](../labs/lab-14.html) is Sampled measurement and zero-order hold. Change one slider at a time, export the series and independently reproduce at least one value from a governing equation. The browser tool is deterministic and offline. It represents a defined educational model, not an approved process simulator. Its chart scale may rescale as parameters change, so compare numerical coordinates rather than visual slope alone.

For a numerical transient, verify step-size sensitivity by recomputing with half the time increment. If a predicted instability disappears merely when the integration time step shrinks, it was likely numerical rather than physical. On the other hand, an unstable physical model may diverge more quickly when more accurately integrated. Preserve initial conditions, parameter units and exact software version in the report so another student can repeat the test.

Practice and oral-examination questions

1. Starting from f_s>2 f_max, identify its state(s), manipulated input(s) and measured output(s). State how you would verify all dimensions.
2. Recalculate the example after a 10% increase of the principal process parameter. What changes in the predicted steady value and in the transient, and why are these not necessarily the same?
3. Propose a sensor or actuator failure that would make an otherwise correct sampling, aliasing, and execution timing model appear wrong. What extra logged variable would distinguish that failure?
4. Explain how Filtering, quantization, and data quality might confound the identification of this lesson’s mechanism and propose a discriminating test.
5. State one safety or operating limit that the simplified f_s>2 f_max relation does not capture. How should an engineer document that omission?

Answer guide and reflection

Question 1 is assessed by correct units and by separating the actual physical inventory from its measured signal; a memorized symbolic expression without those definitions is insufficient. For Question 2, the correct method uses the original equation twice and recomputes the equilibrium or time constant where applicable; simple proportional scaling is not accepted without showing that the equation is linear in the varied coefficient. For Question 3, the measurement or actuator diagnosis must lead to a different predicted trace. For Question 4, a proposed experiment is credible only if it can distinguish two candidate mechanisms. For Question 5, a constraint must be named and its omission explicitly stated.

Takeaway for Digital Controllers, Data, and Alarm Logic. The lesson is complete only after the model can be challenged with a changed initial condition, a disturbance and a non-ideal measurement. A plotted agreement under one nominal set of synthetic conditions is the start of investigation, not its end. The next step is to test whether the assumptions retain their meaning when discrete integration and controller updates is included in the model.

Unit 7, Lesson 2: Discrete integration and controller updates

Unit: Digital Controllers, Data, and Alarm Logic. Physical setting: PLC-based temperature and level control with a digital historian. Reference model: I[k]=I[k-1]+Ki Ts e[k].

Learning outcomes

By the end of this lesson you should be able to (i) explain discrete integration and controller updates in physical terms, (ii) identify the variables and assumptions of I[k]=I[k-1]+Ki Ts e[k], (iii) reproduce and challenge a numerical benchmark, (iv) recognize how measurement, delay and actuator limits modify the model, and (v) design a specific experiment that can distinguish the current model from an alternative interpretation.

Physical reasoning, not just a formula

Digital PI algorithms multiply error by the actual elapsed time. Forward and backward Euler and trapezoidal updates have different numerical characteristics; an implementation should state which it uses.

Consider the unit-wide equipment case: PLC-based temperature and level control with a digital historian. In this setting the same numerical trend may have more than one physical cause. A change observed at the analyzer may reflect a real process response, a changed flow residence time or the instrument dynamics. The analyst must first specify which quantity is manipulated, which is observed and which variables are treated as external disturbances. This is particularly important for discrete integration and controller updates, where confusing a state with a measured output can give a mathematically solvable but physically incorrect model.

The chosen governing expression is I[k]=I[k-1]+Ki Ts e[k]. It is a compact model, not a complete plant specification. Terms on opposite sides of an equality must have compatible units and consistent time bases. When a parameter is estimated from one transient, its result belongs to that operating range unless new data demonstrate transferability. The task is to explain the meaning of the relation before substituting numbers and then to compare its predictions with a checkable boundary condition or steady state.

Working the numerical reference case

Synthetic worked example. With Ki=0.1 s⁻¹, Ts=2 s and e=3 maintained for four samples, the accumulated integral increment equals 0.123*4=2.4 output units.

First transcribe every numerical input into a short table with its physical unit. Second, compute the baseline result without a software solver. Third, identify which output should change and which should remain unchanged when the manipulated input is zero. In this discrete integration and controller updates exercise, the reported figures are not fitted plant data: they are an internally stated teaching benchmark. Any extension to different equipment requires new coefficients and new validation.

A useful sensitivity check is to change the selected coefficient or input by +10% while retaining all other conditions. This is not equivalent to assuming the output must also rise by 10%; exponential responses, ratios, saturation and feedback can change the mapping. Write the original expression, substitute the perturbed value and compare the sign of the result against physical intuition. Next repeat with a −10% change. An asymmetric response is evidence of nonlinearity in the selected mapping, not necessarily a numerical error.

Derivation and model restrictions

If cycle execution is delayed, pretending Ts remained fixed introduces a hidden integration error. To turn a physical inventory into a differential state equation, write accumulation as incoming rate minus outgoing rate plus net production, if relevant. To turn an instrument or actuator into a dynamical element, specify its output, time constant and initial condition. To build a transfer function, take deviations about a steady operating point and verify that the assumed initial deviations are zero. Do not apply frequency-domain algebra blindly to a strongly nonlinear or switching regime.

For discrete integration and controller updates, the limiting checks should include a zero-change input, a long-time equilibrium where one exists and at least one infeasible input that the physical equipment cannot realize. If a parameter has dimensions of reciprocal time, use it only with a time expressed in matching units. If a parameter multiplies a temperature difference, its unit must convert the difference into a heat rate before the quantity is added to the energy balance. These simple checks detect mistakes that a graph renderer will not.

How to identify the model experimentally

Use the previously studied topic Sampling, aliasing, and execution timing as a source of prior knowledge and the next topic Filtering, quantization, and data quality as an independent challenge. Define the reference operating point for discrete integration and controller updates in the process case and record a small safe input sequence, the actual actuator trajectory, the measurement times and instrument range. A step, a pulse and a slow ramp probe different properties, so do not collapse them into one claimed validation. If the fitted output differs from the recorded trace, inspect the timestamp alignment and sensor dynamics before increasing model order.

Where possible, obtain at least two input amplitudes. Normalize the measured output by the change in input and test whether the curves approximately collapse. Failure to collapse can indicate nonlinearity, saturation, temperature-dependent coefficients or changed initial conditions. A second holdout operating point is especially important when the main formula was derived by linearization. Report the residual error and uncertainty rather than relabeling an imperfect approximation as a universal physical law.

Uncertainty, troubleshooting, and design decisions

Failure mode to investigate: A sample time changed during commissioning without retuning gains can destabilize a previously acceptable loop. The first diagnostic question is which conservation law, measurement chain or actuator path the proposed explanation implicates. Reconstruct the actual manipulated-variable history and compare it with what the controller requested; a large difference is a plant-side constraint, not a parameter-estimation error that can be fixed by fitting a more elaborate process transfer function.

Decision from the evidence: Compare difference-equation results against a high-resolution continuous reference across several Ts values. State what the experiment supports and what remains unidentifiable. The decision should include whether the next useful action is to add a sensor, run a second input condition, change an assumed model structure or restrict the admissible operating range. When model outputs could guide actual equipment, independent engineering review and safety analysis are necessary before any implementation.

A challenge to the simplest explanation

Compare the present explanation with Alarm hysteresis, bumpless transfer, and interlocks, which appears elsewhere in this unit. Imagine obtaining a smooth measured response that both models could plausibly produce. Choose one additional measurement that would change differently under the two explanations. Define the expected sign and time scale under each. For discrete integration and controller updates, record not just peak error but also the total actuator travel, disturbances, measurement noise and how the reference signal was generated. This exercise forces you to separate apparent goodness of fit from the ability to predict new conditions.

Numerical implementation and independent checks

The companion [interactive model, Laboratory 07](../labs/lab-07.html) is PI control under a load disturbance. Change one slider at a time, export the series and independently reproduce at least one value from a governing equation. The browser tool is deterministic and offline. It represents a defined educational model, not an approved process simulator. Its chart scale may rescale as parameters change, so compare numerical coordinates rather than visual slope alone.

For a numerical transient, verify step-size sensitivity by recomputing with half the time increment. If a predicted instability disappears merely when the integration time step shrinks, it was likely numerical rather than physical. On the other hand, an unstable physical model may diverge more quickly when more accurately integrated. Preserve initial conditions, parameter units and exact software version in the report so another student can repeat the test.

Practice and oral-examination questions

1. Starting from I[k]=I[k-1]+Ki Ts e[k], identify its state(s), manipulated input(s) and measured output(s). State how you would verify all dimensions.
2. Recalculate the example after a 10% increase of the principal process parameter. What changes in the predicted steady value and in the transient, and why are these not necessarily the same?
3. Propose a sensor or actuator failure that would make an otherwise correct discrete integration and controller updates model appear wrong. What extra logged variable would distinguish that failure?
4. Explain how Alarm hysteresis, bumpless transfer, and interlocks might confound the identification of this lesson’s mechanism and propose a discriminating test.
5. State one safety or operating limit that the simplified I[k]=I[k-1]+Ki Ts e[k] relation does not capture. How should an engineer document that omission?

Answer guide and reflection

Question 1 is assessed by correct units and by separating the actual physical inventory from its measured signal; a memorized symbolic expression without those definitions is insufficient. For Question 2, the correct method uses the original equation twice and recomputes the equilibrium or time constant where applicable; simple proportional scaling is not accepted without showing that the equation is linear in the varied coefficient. For Question 3, the measurement or actuator diagnosis must lead to a different predicted trace. For Question 4, a proposed experiment is credible only if it can distinguish two candidate mechanisms. For Question 5, a constraint must be named and its omission explicitly stated.

Takeaway for Digital Controllers, Data, and Alarm Logic. The lesson is complete only after the model can be challenged with a changed initial condition, a disturbance and a non-ideal measurement. A plotted agreement under one nominal set of synthetic conditions is the start of investigation, not its end. The next step is to test whether the assumptions retain their meaning when filtering, quantization, and data quality is included in the model.

Unit 7, Lesson 3: Filtering, quantization, and data quality

Unit: Digital Controllers, Data, and Alarm Logic. Physical setting: PLC-based temperature and level control with a digital historian. Reference model: y_f[k]=alpha y_f[k-1]+(1-alpha)y[k].

Learning outcomes

By the end of this lesson you should be able to (i) explain filtering, quantization, and data quality in physical terms, (ii) identify the variables and assumptions of y_f[k]=alpha y_f[k-1]+(1-alpha)y[k], (iii) reproduce and challenge a numerical benchmark, (iv) recognize how measurement, delay and actuator limits modify the model, and (v) design a specific experiment that can distinguish the current model from an alternative interpretation.

Physical reasoning, not just a formula

A digital exponential filter moderates noise but creates latency, while sensor or ADC quantization causes staircase measurements. Missing or bad-quality samples must be handled by policy, not silently treated as valid zeros.

Consider the unit-wide equipment case: PLC-based temperature and level control with a digital historian. In this setting the same numerical trend may have more than one physical cause. A change observed at the analyzer may reflect a real process response, a changed flow residence time or the instrument dynamics. The analyst must first specify which quantity is manipulated, which is observed and which variables are treated as external disturbances. This is particularly important for filtering, quantization, and data quality, where confusing a state with a measured output can give a mathematically solvable but physically incorrect model.

The chosen governing expression is y_f[k]=alpha y_f[k-1]+(1-alpha)y[k]. It is a compact model, not a complete plant specification. Terms on opposite sides of an equality must have compatible units and consistent time bases. When a parameter is estimated from one transient, its result belongs to that operating range unless new data demonstrate transferability. The task is to explain the meaning of the relation before substituting numbers and then to compare its predictions with a checkable boundary condition or steady state.

Working the numerical reference case

Synthetic worked example. With alpha=0.8, previous filtered value 20 and new sample 30, updated output is 0.820+0.230=22. A one-count resolution on 0–100 over 10 bits is roughly 0.0977 units.

First transcribe every numerical input into a short table with its physical unit. Second, compute the baseline result without a software solver. Third, identify which output should change and which should remain unchanged when the manipulated input is zero. In this filtering, quantization, and data quality exercise, the reported figures are not fitted plant data: they are an internally stated teaching benchmark. Any extension to different equipment requires new coefficients and new validation.

A useful sensitivity check is to change the selected coefficient or input by +10% while retaining all other conditions. This is not equivalent to assuming the output must also rise by 10%; exponential responses, ratios, saturation and feedback can change the mapping. Write the original expression, substitute the perturbed value and compare the sign of the result against physical intuition. Next repeat with a −10% change. An asymmetric response is evidence of nonlinearity in the selected mapping, not necessarily a numerical error.

Derivation and model restrictions

Filter alpha is tied to sampling interval; a fixed alpha with changed Ts changes its real time constant. To turn a physical inventory into a differential state equation, write accumulation as incoming rate minus outgoing rate plus net production, if relevant. To turn an instrument or actuator into a dynamical element, specify its output, time constant and initial condition. To build a transfer function, take deviations about a steady operating point and verify that the assumed initial deviations are zero. Do not apply frequency-domain algebra blindly to a strongly nonlinear or switching regime.

For filtering, quantization, and data quality, the limiting checks should include a zero-change input, a long-time equilibrium where one exists and at least one infeasible input that the physical equipment cannot realize. If a parameter has dimensions of reciprocal time, use it only with a time expressed in matching units. If a parameter multiplies a temperature difference, its unit must convert the difference into a heat rate before the quantity is added to the energy balance. These simple checks detect mistakes that a graph renderer will not.

How to identify the model experimentally

Use the previously studied topic Discrete integration and controller updates as a source of prior knowledge and the next topic Alarm hysteresis, bumpless transfer, and interlocks as an independent challenge. Define the reference operating point for filtering, quantization, and data quality in the process case and record a small safe input sequence, the actual actuator trajectory, the measurement times and instrument range. A step, a pulse and a slow ramp probe different properties, so do not collapse them into one claimed validation. If the fitted output differs from the recorded trace, inspect the timestamp alignment and sensor dynamics before increasing model order.

Where possible, obtain at least two input amplitudes. Normalize the measured output by the change in input and test whether the curves approximately collapse. Failure to collapse can indicate nonlinearity, saturation, temperature-dependent coefficients or changed initial conditions. A second holdout operating point is especially important when the main formula was derived by linearization. Report the residual error and uncertainty rather than relabeling an imperfect approximation as a universal physical law.

Uncertainty, troubleshooting, and design decisions

Failure mode to investigate: More filtering can reduce visible noise yet magnify closed-loop overshoot by slowing feedback. The first diagnostic question is which conservation law, measurement chain or actuator path the proposed explanation implicates. Reconstruct the actual manipulated-variable history and compare it with what the controller requested; a large difference is a plant-side constraint, not a parameter-estimation error that can be fixed by fitting a more elaborate process transfer function.

Decision from the evidence: Vary sample time and alpha while plotting delay, noise attenuation and digitization staircases. State what the experiment supports and what remains unidentifiable. The decision should include whether the next useful action is to add a sensor, run a second input condition, change an assumed model structure or restrict the admissible operating range. When model outputs could guide actual equipment, independent engineering review and safety analysis are necessary before any implementation.

A challenge to the simplest explanation

Compare the present explanation with Sampling, aliasing, and execution timing, which appears elsewhere in this unit. Imagine obtaining a smooth measured response that both models could plausibly produce. Choose one additional measurement that would change differently under the two explanations. Define the expected sign and time scale under each. For filtering, quantization, and data quality, record not just peak error but also the total actuator travel, disturbances, measurement noise and how the reference signal was generated. This exercise forces you to separate apparent goodness of fit from the ability to predict new conditions.

Numerical implementation and independent checks

The companion [interactive model, Laboratory 15](../labs/lab-15.html) is Digital low-pass filtering. Change one slider at a time, export the series and independently reproduce at least one value from a governing equation. The browser tool is deterministic and offline. It represents a defined educational model, not an approved process simulator. Its chart scale may rescale as parameters change, so compare numerical coordinates rather than visual slope alone.

For a numerical transient, verify step-size sensitivity by recomputing with half the time increment. If a predicted instability disappears merely when the integration time step shrinks, it was likely numerical rather than physical. On the other hand, an unstable physical model may diverge more quickly when more accurately integrated. Preserve initial conditions, parameter units and exact software version in the report so another student can repeat the test.

Practice and oral-examination questions

1. Starting from y_f[k]=alpha y_f[k-1]+(1-alpha)y[k], identify its state(s), manipulated input(s) and measured output(s). State how you would verify all dimensions.
2. Recalculate the example after a 10% increase of the principal process parameter. What changes in the predicted steady value and in the transient, and why are these not necessarily the same?
3. Propose a sensor or actuator failure that would make an otherwise correct filtering, quantization, and data quality model appear wrong. What extra logged variable would distinguish that failure?
4. Explain how Sampling, aliasing, and execution timing might confound the identification of this lesson’s mechanism and propose a discriminating test.
5. State one safety or operating limit that the simplified y_f[k]=alpha y_f[k-1]+(1-alpha)y[k] relation does not capture. How should an engineer document that omission?

Answer guide and reflection

Question 1 is assessed by correct units and by separating the actual physical inventory from its measured signal; a memorized symbolic expression without those definitions is insufficient. For Question 2, the correct method uses the original equation twice and recomputes the equilibrium or time constant where applicable; simple proportional scaling is not accepted without showing that the equation is linear in the varied coefficient. For Question 3, the measurement or actuator diagnosis must lead to a different predicted trace. For Question 4, a proposed experiment is credible only if it can distinguish two candidate mechanisms. For Question 5, a constraint must be named and its omission explicitly stated.

Takeaway for Digital Controllers, Data, and Alarm Logic. The lesson is complete only after the model can be challenged with a changed initial condition, a disturbance and a non-ideal measurement. A plotted agreement under one nominal set of synthetic conditions is the start of investigation, not its end. The next step is to test whether the assumptions retain their meaning when alarm hysteresis, bumpless transfer, and interlocks is included in the model.

Unit 7, Lesson 4: Alarm hysteresis, bumpless transfer, and interlocks

Unit: Digital Controllers, Data, and Alarm Logic. Physical setting: PLC-based temperature and level control with a digital historian. Reference model: alarm_on if y>=H; alarm_off if y<=L.

Learning outcomes

By the end of this lesson you should be able to (i) explain alarm hysteresis, bumpless transfer, and interlocks in physical terms, (ii) identify the variables and assumptions of alarm_on if y>=H; alarm_off if y<=L, (iii) reproduce and challenge a numerical benchmark, (iv) recognize how measurement, delay and actuator limits modify the model, and (v) design a specific experiment that can distinguish the current model from an alternative interpretation.

Physical reasoning, not just a formula

Alarms inform operators, while protective interlocks act under defined conditions and should be engineered independently of ordinary regulation. Hysteresis and delay can prevent chattering but cannot replace hazard review.

Consider the unit-wide equipment case: PLC-based temperature and level control with a digital historian. In this setting the same numerical trend may have more than one physical cause. A change observed at the analyzer may reflect a real process response, a changed flow residence time or the instrument dynamics. The analyst must first specify which quantity is manipulated, which is observed and which variables are treated as external disturbances. This is particularly important for alarm hysteresis, bumpless transfer, and interlocks, where confusing a state with a measured output can give a mathematically solvable but physically incorrect model.

The chosen governing expression is alarm_on if y>=H; alarm_off if y<=L. It is a compact model, not a complete plant specification. Terms on opposite sides of an equality must have compatible units and consistent time bases. When a parameter is estimated from one transient, its result belongs to that operating range unless new data demonstrate transferability. The task is to explain the meaning of the relation before substituting numbers and then to compare its predictions with a checkable boundary condition or steady state.

Working the numerical reference case

Synthetic worked example. An illustrative high-temperature alarm raises at 90 °C and clears at 87 °C. A trace 89→91→89→88→86 °C triggers once and clears only at 86 °C.

First transcribe every numerical input into a short table with its physical unit. Second, compute the baseline result without a software solver. Third, identify which output should change and which should remain unchanged when the manipulated input is zero. In this alarm hysteresis, bumpless transfer, and interlocks exercise, the reported figures are not fitted plant data: they are an internally stated teaching benchmark. Any extension to different equipment requires new coefficients and new validation.

A useful sensitivity check is to change the selected coefficient or input by +10% while retaining all other conditions. This is not equivalent to assuming the output must also rise by 10%; exponential responses, ratios, saturation and feedback can change the mapping. Write the original expression, substitute the perturbed value and compare the sign of the result against physical intuition. Next repeat with a −10% change. An asymmetric response is evidence of nonlinearity in the selected mapping, not necessarily a numerical error.

Derivation and model restrictions

Thresholds are educational examples, not recommended setpoints for any process; acceptable thresholds require site-specific risk analysis. To turn a physical inventory into a differential state equation, write accumulation as incoming rate minus outgoing rate plus net production, if relevant. To turn an instrument or actuator into a dynamical element, specify its output, time constant and initial condition. To build a transfer function, take deviations about a steady operating point and verify that the assumed initial deviations are zero. Do not apply frequency-domain algebra blindly to a strongly nonlinear or switching regime.

For alarm hysteresis, bumpless transfer, and interlocks, the limiting checks should include a zero-change input, a long-time equilibrium where one exists and at least one infeasible input that the physical equipment cannot realize. If a parameter has dimensions of reciprocal time, use it only with a time expressed in matching units. If a parameter multiplies a temperature difference, its unit must convert the difference into a heat rate before the quantity is added to the energy balance. These simple checks detect mistakes that a graph renderer will not.

How to identify the model experimentally

Use the previously studied topic Filtering, quantization, and data quality as a source of prior knowledge and the next topic Alarm hysteresis, bumpless transfer, and interlocks as an independent challenge. Define the reference operating point for alarm hysteresis, bumpless transfer, and interlocks in the process case and record a small safe input sequence, the actual actuator trajectory, the measurement times and instrument range. A step, a pulse and a slow ramp probe different properties, so do not collapse them into one claimed validation. If the fitted output differs from the recorded trace, inspect the timestamp alignment and sensor dynamics before increasing model order.

Where possible, obtain at least two input amplitudes. Normalize the measured output by the change in input and test whether the curves approximately collapse. Failure to collapse can indicate nonlinearity, saturation, temperature-dependent coefficients or changed initial conditions. A second holdout operating point is especially important when the main formula was derived by linearization. Report the residual error and uncertainty rather than relabeling an imperfect approximation as a universal physical law.

Uncertainty, troubleshooting, and design decisions

Failure mode to investigate: Without hysteresis, noisy measurements around 90 °C can generate dozens of nuisance alarms in a few minutes. The first diagnostic question is which conservation law, measurement chain or actuator path the proposed explanation implicates. Reconstruct the actual manipulated-variable history and compare it with what the controller requested; a large difference is a plant-side constraint, not a parameter-estimation error that can be fixed by fitting a more elaborate process transfer function.

Decision from the evidence: Describe manual-to-auto transfer with matched integral state and test sensor disconnection, stuck values and restored communications. State what the experiment supports and what remains unidentifiable. The decision should include whether the next useful action is to add a sensor, run a second input condition, change an assumed model structure or restrict the admissible operating range. When model outputs could guide actual equipment, independent engineering review and safety analysis are necessary before any implementation.

A challenge to the simplest explanation

Compare the present explanation with Discrete integration and controller updates, which appears elsewhere in this unit. Imagine obtaining a smooth measured response that both models could plausibly produce. Choose one additional measurement that would change differently under the two explanations. Define the expected sign and time scale under each. For alarm hysteresis, bumpless transfer, and interlocks, record not just peak error but also the total actuator travel, disturbances, measurement noise and how the reference signal was generated. This exercise forces you to separate apparent goodness of fit from the ability to predict new conditions.

Numerical implementation and independent checks

The companion [interactive model, Laboratory 16](../labs/lab-16.html) is Alarm hysteresis and switching. Change one slider at a time, export the series and independently reproduce at least one value from a governing equation. The browser tool is deterministic and offline. It represents a defined educational model, not an approved process simulator. Its chart scale may rescale as parameters change, so compare numerical coordinates rather than visual slope alone.

For a numerical transient, verify step-size sensitivity by recomputing with half the time increment. If a predicted instability disappears merely when the integration time step shrinks, it was likely numerical rather than physical. On the other hand, an unstable physical model may diverge more quickly when more accurately integrated. Preserve initial conditions, parameter units and exact software version in the report so another student can repeat the test.

Practice and oral-examination questions

1. Starting from alarm_on if y>=H; alarm_off if y<=L, identify its state(s), manipulated input(s) and measured output(s). State how you would verify all dimensions.
2. Recalculate the example after a 10% increase of the principal process parameter. What changes in the predicted steady value and in the transient, and why are these not necessarily the same?
3. Propose a sensor or actuator failure that would make an otherwise correct alarm hysteresis, bumpless transfer, and interlocks model appear wrong. What extra logged variable would distinguish that failure?
4. Explain how Discrete integration and controller updates might confound the identification of this lesson’s mechanism and propose a discriminating test.
5. State one safety or operating limit that the simplified alarm_on if y>=H; alarm_off if y<=L relation does not capture. How should an engineer document that omission?

Answer guide and reflection

Question 1 is assessed by correct units and by separating the actual physical inventory from its measured signal; a memorized symbolic expression without those definitions is insufficient. For Question 2, the correct method uses the original equation twice and recomputes the equilibrium or time constant where applicable; simple proportional scaling is not accepted without showing that the equation is linear in the varied coefficient. For Question 3, the measurement or actuator diagnosis must lead to a different predicted trace. For Question 4, a proposed experiment is credible only if it can distinguish two candidate mechanisms. For Question 5, a constraint must be named and its omission explicitly stated.

Takeaway for Digital Controllers, Data, and Alarm Logic. The lesson is complete only after the model can be challenged with a changed initial condition, a disturbance and a non-ideal measurement. A plotted agreement under one nominal set of synthetic conditions is the start of investigation, not its end. The next step is to test whether the assumptions retain their meaning when alarm hysteresis, bumpless transfer, and interlocks is included in the model.

Unit 7 applied project: Digital Controllers, Data, and Alarm Logic

Decision problem

You are an analyst assigned to PLC-based temperature and level control with a digital historian. A proposed controller or model change claims better performance. Your job is not to accept the claim but to define a reproducible counterfactual using the physical assumptions of this unit, quantify expected benefits, identify failure modes and explain what evidence would change the decision. Use deliberately synthesized teaching data unless real data are supplied with permission.

Required technical workflow

Stage 1 — Sampling, aliasing, and execution timing. Use f_s>2 f_max to predict a measurable effect. Compare the same physical signal at 1, 5 and 20 samples per second and calculate apparent frequency. Explain how an operator trend that aliases oscillation can falsely suggest the equipment is changing slowly. Use the example conditions in the lesson only as a preliminary reference case; the final project should vary at least two different operating conditions so that conclusions do not depend on one curve.

Stage 2 — Discrete integration and controller updates. Use I[k]=I[k-1]+Ki Ts e[k] to predict a measurable effect. Compare difference-equation results against a high-resolution continuous reference across several Ts values. Explain how a sample time changed during commissioning without retuning gains can destabilize a previously acceptable loop. Use the example conditions in the lesson only as a preliminary reference case; the final project should vary at least two different operating conditions so that conclusions do not depend on one curve.

Stage 3 — Filtering, quantization, and data quality. Use y_f[k]=alpha y_f[k-1]+(1-alpha)y[k] to predict a measurable effect. Vary sample time and alpha while plotting delay, noise attenuation and digitization staircases. Explain how more filtering can reduce visible noise yet magnify closed-loop overshoot by slowing feedback. Use the example conditions in the lesson only as a preliminary reference case; the final project should vary at least two different operating conditions so that conclusions do not depend on one curve.

Stage 4 — Alarm hysteresis, bumpless transfer, and interlocks. Use alarm_on if y>=H; alarm_off if y<=L to predict a measurable effect. Describe manual-to-auto transfer with matched integral state and test sensor disconnection, stuck values and restored communications. Explain how without hysteresis, noisy measurements around 90 °c can generate dozens of nuisance alarms in a few minutes. Use the example conditions in the lesson only as a preliminary reference case; the final project should vary at least two different operating conditions so that conclusions do not depend on one curve.

Evidence package

Provide a diagram with input, state, output and disturbance labels; a parameter dictionary with units and evidence status; the precise model equations; a csv or spreadsheet of numerical results; two labelled charts; and a structured comparison of nominal and perturbed cases. The input changes must be safe in the synthetic scenario and no physical equipment actuation is authorized by this assignment.

Model comparison and failure analysis

Compare a simple low-order model with a second model that adds one relevant limitation from sampled measurements; numerical integration; quantization/filtering; alarm management. Keep the initial conditions and disturbance realization consistent. Calculate peak deviation, time to regain a chosen tolerance, integrated absolute error and the total excursion of the manipulated input when a controller is present. Explain any model disagreement using equation terms, not subjective plot descriptions.

Deliverables and evaluation

Submit a 3–5 page technical memo, runnable offline files or documented equations, a small data table and an audit statement. The recommended change must be limited to the modeled conditions. State at least two uncertainties, one additional validation experiment, and any safety question outside the authority of a classroom model. The grading priority is transparent reasoning and reproducibility, not a claimed industrial optimum.

Unit 7 assessment: Digital Controllers, Data, and Alarm Logic

Assessment type: open-book problem set. Total: 16 tasks plus one integrative evaluation. Show reasoning, units, model assumptions, and at least one verification check for each numerical answer. Data are illustrative rather than plant measurements.

Part 1: Sampling, aliasing, and execution timing

Question 1. Derive or interpret f_s>2 f_max. Define its coefficient units, the state variable, its initial condition and the operating point if linearized.

Question 2. Using the synthetic benchmark described in the corresponding lesson (A 0.8 Hz sinusoid sampled at 1 Hz appears as a 0.2 Hz alias in uniformly sampled data; sampling has erased the original frequency information.), repeat the arithmetic, then alter one explicitly stated input by +15% and explain the sign of the difference.

Question 3. Challenge the approximation: Nyquist is a theoretical limit for perfect bandlimited signals; control design typically needs a much faster sample rate than the bare factor of two. Describe an experiment, instrument placement and observation that would invalidate this assumption.

Question 4. Troubleshoot the known risk: An operator trend that aliases oscillation can falsely suggest the equipment is changing slowly. Propose a controlled test and a quantitative metric by which two possible causes can be separated.

Part 2: Discrete integration and controller updates

Question 5. Derive or interpret I[k]=I[k-1]+Ki Ts e[k]. Define its coefficient units, the state variable, its initial condition and the operating point if linearized.

Question 6. Using the synthetic benchmark described in the corresponding lesson (With Ki=0.1 s⁻¹, Ts=2 s and e=3 maintained for four samples, the accumulated integral increment equals 0.123*4=2.4 output units.), repeat the arithmetic, then alter one explicitly stated input by +15% and explain the sign of the difference.

Question 7. Challenge the approximation: If cycle execution is delayed, pretending Ts remained fixed introduces a hidden integration error. Describe an experiment, instrument placement and observation that would invalidate this assumption.

Question 8. Troubleshoot the known risk: A sample time changed during commissioning without retuning gains can destabilize a previously acceptable loop. Propose a controlled test and a quantitative metric by which two possible causes can be separated.

Part 3: Filtering, quantization, and data quality

Question 9. Derive or interpret y_f[k]=alpha y_f[k-1]+(1-alpha)y[k]. Define its coefficient units, the state variable, its initial condition and the operating point if linearized.

Question 10. Using the synthetic benchmark described in the corresponding lesson (With alpha=0.8, previous filtered value 20 and new sample 30, updated output is 0.820+0.230=22. A one-count resolution on 0–100 over 10 bits is roughly 0.0977 units.), repeat the arithmetic, then alter one explicitly stated input by +15% and explain the sign of the difference.

Question 11. Challenge the approximation: Filter alpha is tied to sampling interval; a fixed alpha with changed Ts changes its real time constant. Describe an experiment, instrument placement and observation that would invalidate this assumption.

Question 12. Troubleshoot the known risk: More filtering can reduce visible noise yet magnify closed-loop overshoot by slowing feedback. Propose a controlled test and a quantitative metric by which two possible causes can be separated.

Part 4: Alarm hysteresis, bumpless transfer, and interlocks

Question 13. Derive or interpret alarm_on if y>=H; alarm_off if y<=L. Define its coefficient units, the state variable, its initial condition and the operating point if linearized.

Question 14. Using the synthetic benchmark described in the corresponding lesson (An illustrative high-temperature alarm raises at 90 °C and clears at 87 °C. A trace 89→91→89→88→86 °C triggers once and clears only at 86 °C.), repeat the arithmetic, then alter one explicitly stated input by +15% and explain the sign of the difference.

Question 15. Challenge the approximation: Thresholds are educational examples, not recommended setpoints for any process; acceptable thresholds require site-specific risk analysis. Describe an experiment, instrument placement and observation that would invalidate this assumption.

Question 16. Troubleshoot the known risk: Without hysteresis, noisy measurements around 90 °C can generate dozens of nuisance alarms in a few minutes. Propose a controlled test and a quantitative metric by which two possible causes can be separated.

Integrative design problem

For the case of PLC-based temperature and level control with a digital historian, select two lessons from this unit and build a consistent process-and-measurement model. Include one realistic actuator bound, one sensor imperfection and one disturbance. Explain the trade-off between fast response and robust operation. Present at least one table of simulated values and one limiting-value check. Label synthetic data distinctly from measured data.

Marking rubric

• 30% governing equations and physical interpretation; 20% internally consistent numerical reasoning; 20% model/assumption checks; 20% alternative hypothesis and experiment design; 10% clear figures, units, and communication.

Examiner notes

Answers are not unique. A valid derivation states a control volume, assumptions, and dimensions before substituting numbers. Parameter changes should be recomputed using the actual model rather than asserted by proportion. If the student finds that the available inputs do not identify a required quantity, a defensible bound and an appropriate additional measurement earn more credit than an invented precise number.