PID Controller Sizing and Loop Tuning

Scope: proportional offset; integral accumulation; derivative filtering; robust tuning and anti-windup. Cross-cutting case: heated-tank temperature regulation with actuator limits.

What this unit builds

Digital control does not remove continuous physics. The code operates at discrete instants while inventories, flows and heat transfer proceed continuously. Numerical integration, filtering and saturation are part of the implemented controller and may change the predicted behavior. Modeling and testing must specify the actual sampling and execution assumptions.

Lesson sequence

1. [Proportional action and steady offset](../lessons/unit-04-lesson-01.html) — Proportional control corrects error immediately but typically leaves a nonzero setpoint offset for processes with finite static gain and load. Higher gain reduces offset while increasing actuator movement and possibly reducing stability margins. Governing relation: `u=Kp(r-y)`.

2. [Integral action, bias and anti-windup](../lessons/unit-04-lesson-02.html) — Integral action accumulates sustained control error and can eliminate steady-state offset for stabilizable plants. Without anti-windup, an actuator held at its stop causes integral state to grow until recovery becomes slow or oscillatory. Governing relation: `u=Kp e+Ki int(e dt)`.

3. [Derivative action, filters, and noise](../lessons/unit-04-lesson-03.html) — Derivative action can anticipate changes in measured output but strongly amplifies high-frequency measurement noise. A filtered derivative on measurement avoids the setpoint kick from differentiating a step in reference. Governing relation: `u_D=-Kd dy_f/dt`.

4. [Model-based tuning and robustness](../lessons/unit-04-lesson-04.html) — Internal-model-control-inspired PI tuning exposes an explicit closed-loop speed target lambda. Choosing very small lambda relative to process delay and model uncertainty can demand infeasible valve movement. Governing relation: `Kc = tau/[K(lambda+theta)]`.

Model connections and physical interpretation

Proportional action and steady offset

Proportional control corrects error immediately but typically leaves a nonzero setpoint offset for processes with finite static gain and load. Higher gain reduces offset while increasing actuator movement and possibly reducing stability margins. The offset result assumes unity feedback, no bias term, a stable loop and no actuator saturation. The calculation that anchors this lesson is deliberately synthetic: For plant static gain 2 and Kp=3, unit setpoint reference yields y_ss=6/7=0.857; the remaining error is 0.143 without integral action. 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.

Integral action, bias and anti-windup

Integral action accumulates sustained control error and can eliminate steady-state offset for stabilizable plants. Without anti-windup, an actuator held at its stop causes integral state to grow until recovery becomes slow or oscillatory. A PI algorithm must specify integration units, discrete sample period, manual/auto transitions and saturation logic. The calculation that anchors this lesson is deliberately synthetic: If Ki=0.20 actuator-units/(error-unit·s) and constant e=2 persists for 15 s, the integral contribution grows by 0.20215=6 actuator units before any limit is applied. 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.

Derivative action, filters, and noise

Derivative action can anticipate changes in measured output but strongly amplifies high-frequency measurement noise. A filtered derivative on measurement avoids the setpoint kick from differentiating a step in reference. Digital differentiation is a finite-difference approximation; the derivative filter adds lag and needs a documented bandwidth. The calculation that anchors this lesson is deliberately synthetic: A sensor fluctuation of 0.10 K over 0.02 s creates an unfiltered slope estimate of 5 K/s. With Kd=3 actuator-units·s/K, the instantaneous derivative contribution can reach 15 actuator 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.

Model-based tuning and robustness

Internal-model-control-inspired PI tuning exposes an explicit closed-loop speed target lambda. Choosing very small lambda relative to process delay and model uncertainty can demand infeasible valve movement. This is one illustrative approximate tuning relation, not a universal optimum. Process type, objective and constraints matter. The calculation that anchors this lesson is deliberately synthetic: With K=2, tau=60 s, theta=10 s, and lambda=40 s, Kc=60/[2*(40+10)]=0.60; a common FOPDT PI rule sets Ti≈tau=60 s for this simplified case. 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 heated-tank temperature regulation with actuator limits. 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 Proportional action and steady offset, state the terms of u=Kp(r-y), choose a limiting or equilibrium case, and explain when it would be invalid.
2. For Integral action, bias and anti-windup, state the terms of u=Kp e+Ki int(e dt), choose a limiting or equilibrium case, and explain when it would be invalid.
3. For Derivative action, filters, and noise, state the terms of u_D=-Kd dy_f/dt, choose a limiting or equilibrium case, and explain when it would be invalid.
4. For Model-based tuning and robustness, state the terms of Kc = tau/[K(lambda+theta)], choose a limiting or equilibrium case, and explain when it would be invalid.

Continue to the [Unit 4 assessment](../assessments/unit-04.html) or [Unit 4 project](../projects/unit-04.html).

Unit 4, Lesson 1: Proportional action and steady offset

Unit: PID Control and Tuning. Physical setting: heated-tank temperature regulation with actuator limits. Reference model: u=Kp(r-y).

Learning outcomes

By the end of this lesson you should be able to (i) explain proportional action and steady offset in physical terms, (ii) identify the variables and assumptions of u=Kp(r-y), (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

Proportional control corrects error immediately but typically leaves a nonzero setpoint offset for processes with finite static gain and load. Higher gain reduces offset while increasing actuator movement and possibly reducing stability margins.

Consider the unit-wide equipment case: heated-tank temperature regulation with actuator limits. 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 proportional action and steady offset, where confusing a state with a measured output can give a mathematically solvable but physically incorrect model.

The chosen governing expression is u=Kp(r-y). 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. For plant static gain 2 and Kp=3, unit setpoint reference yields y_ss=6/7=0.857; the remaining error is 0.143 without integral action.

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 proportional action and steady offset 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

The offset result assumes unity feedback, no bias term, a stable loop and no actuator saturation. 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 proportional action and steady offset, 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 Proportional action and steady offset as a source of prior knowledge and the next topic Integral action, bias and anti-windup as an independent challenge. Define the reference operating point for proportional action and steady offset 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 noisy instrument can convert high proportional gain into rapid valve cycling and shortened actuator life. 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: Increase gain in controlled increments, plotting response time and final offset alongside manipulated-variable excursion. 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 Derivative action, filters, and noise, 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 proportional action and steady offset, 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 06](../labs/lab-06.html) is Proportional feedback and offset. 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 u=Kp(r-y), 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 proportional action and steady offset model appear wrong. What extra logged variable would distinguish that failure?
4. Explain how Derivative action, filters, and noise might confound the identification of this lesson’s mechanism and propose a discriminating test.
5. State one safety or operating limit that the simplified u=Kp(r-y) 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 PID Control and Tuning. 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 integral action, bias and anti-windup is included in the model.

Unit 4, Lesson 2: Integral action, bias and anti-windup

Unit: PID Control and Tuning. Physical setting: heated-tank temperature regulation with actuator limits. Reference model: u=Kp e+Ki int(e dt).

Learning outcomes

By the end of this lesson you should be able to (i) explain integral action, bias and anti-windup in physical terms, (ii) identify the variables and assumptions of u=Kp e+Ki int(e dt), (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

Integral action accumulates sustained control error and can eliminate steady-state offset for stabilizable plants. Without anti-windup, an actuator held at its stop causes integral state to grow until recovery becomes slow or oscillatory.

Consider the unit-wide equipment case: heated-tank temperature regulation with actuator limits. 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 integral action, bias and anti-windup, where confusing a state with a measured output can give a mathematically solvable but physically incorrect model.

The chosen governing expression is u=Kp e+Ki int(e dt). 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. If Ki=0.20 actuator-units/(error-unit·s) and constant e=2 persists for 15 s, the integral contribution grows by 0.20215=6 actuator units before any limit is applied.

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 integral action, bias and anti-windup 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

A PI algorithm must specify integration units, discrete sample period, manual/auto transitions and saturation logic. 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 integral action, bias and anti-windup, 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 Proportional action and steady offset as a source of prior knowledge and the next topic Derivative action, filters, and noise as an independent challenge. Define the reference operating point for integral action, bias and anti-windup 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 integral term that keeps integrating while a valve is fully open delays correction after disturbance reversal. 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 clamped integration with unconstrained integration during a large setpoint change. 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 Model-based tuning and robustness, 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 integral action, bias and anti-windup, 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 u=Kp e+Ki int(e dt), 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 integral action, bias and anti-windup model appear wrong. What extra logged variable would distinguish that failure?
4. Explain how Model-based tuning and robustness might confound the identification of this lesson’s mechanism and propose a discriminating test.
5. State one safety or operating limit that the simplified u=Kp e+Ki int(e dt) 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 PID Control and Tuning. 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 derivative action, filters, and noise is included in the model.

Unit 4, Lesson 3: Derivative action, filters, and noise

Unit: PID Control and Tuning. Physical setting: heated-tank temperature regulation with actuator limits. Reference model: u_D=-Kd dy_f/dt.

Learning outcomes

By the end of this lesson you should be able to (i) explain derivative action, filters, and noise in physical terms, (ii) identify the variables and assumptions of u_D=-Kd dy_f/dt, (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

Derivative action can anticipate changes in measured output but strongly amplifies high-frequency measurement noise. A filtered derivative on measurement avoids the setpoint kick from differentiating a step in reference.

Consider the unit-wide equipment case: heated-tank temperature regulation with actuator limits. 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 derivative action, filters, and noise, where confusing a state with a measured output can give a mathematically solvable but physically incorrect model.

The chosen governing expression is u_D=-Kd dy_f/dt. 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 sensor fluctuation of 0.10 K over 0.02 s creates an unfiltered slope estimate of 5 K/s. With Kd=3 actuator-units·s/K, the instantaneous derivative contribution can reach 15 actuator 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 derivative action, filters, and noise 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

Digital differentiation is a finite-difference approximation; the derivative filter adds lag and needs a documented bandwidth. 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 derivative action, filters, and noise, 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 Integral action, bias and anti-windup as a source of prior knowledge and the next topic Model-based tuning and robustness as an independent challenge. Define the reference operating point for derivative action, filters, and noise 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 ideal continuous derivative in a textbook is physically unrealizable in noisy instrumentation. 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: Repeat tuning with the same process model but increasing noise to expose why derivative filtering matters. 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 Proportional action and steady offset, 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 derivative action, filters, and noise, 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 08](../labs/lab-08.html) is PID with actuator saturation and anti-windup. 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 u_D=-Kd dy_f/dt, 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 derivative action, filters, and noise model appear wrong. What extra logged variable would distinguish that failure?
4. Explain how Proportional action and steady offset might confound the identification of this lesson’s mechanism and propose a discriminating test.
5. State one safety or operating limit that the simplified u_D=-Kd dy_f/dt 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 PID Control and Tuning. 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 model-based tuning and robustness is included in the model.

Unit 4, Lesson 4: Model-based tuning and robustness

Unit: PID Control and Tuning. Physical setting: heated-tank temperature regulation with actuator limits. Reference model: Kc = tau/[K(lambda+theta)].

Learning outcomes

By the end of this lesson you should be able to (i) explain model-based tuning and robustness in physical terms, (ii) identify the variables and assumptions of Kc = tau/[K(lambda+theta)], (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

Internal-model-control-inspired PI tuning exposes an explicit closed-loop speed target lambda. Choosing very small lambda relative to process delay and model uncertainty can demand infeasible valve movement.

Consider the unit-wide equipment case: heated-tank temperature regulation with actuator limits. 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 model-based tuning and robustness, where confusing a state with a measured output can give a mathematically solvable but physically incorrect model.

The chosen governing expression is Kc = tau/[K(lambda+theta)]. 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 K=2, tau=60 s, theta=10 s, and lambda=40 s, Kc=60/[2*(40+10)]=0.60; a common FOPDT PI rule sets Ti≈tau=60 s for this simplified case.

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 model-based tuning and robustness 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

This is one illustrative approximate tuning relation, not a universal optimum. Process type, objective and constraints matter. 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 model-based tuning and robustness, 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 Derivative action, filters, and noise as a source of prior knowledge and the next topic Model-based tuning and robustness as an independent challenge. Define the reference operating point for model-based tuning and robustness 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 apparently faster closed-loop design can fail when delayed measurement or valve saturation invalidates the nominal process model. 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: Test tuning at ±20% gain, lag and delay variation and evaluate integral absolute error and valve travel. 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 Integral action, bias and anti-windup, 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 model-based tuning and robustness, 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 Kc = tau/[K(lambda+theta)], 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 model-based tuning and robustness model appear wrong. What extra logged variable would distinguish that failure?
4. Explain how Integral action, bias and anti-windup might confound the identification of this lesson’s mechanism and propose a discriminating test.
5. State one safety or operating limit that the simplified Kc = tau/[K(lambda+theta)] 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 PID Control and Tuning. 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 model-based tuning and robustness is included in the model.

Unit 4 applied project: PID Control and Tuning

Decision problem

You are an analyst assigned to heated-tank temperature regulation with actuator limits. 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 — Proportional action and steady offset. Use u=Kp(r-y) to predict a measurable effect. Increase gain in controlled increments, plotting response time and final offset alongside manipulated-variable excursion. Explain how a noisy instrument can convert high proportional gain into rapid valve cycling and shortened actuator life. 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 — Integral action, bias and anti-windup. Use u=Kp e+Ki int(e dt) to predict a measurable effect. Compare clamped integration with unconstrained integration during a large setpoint change. Explain how an integral term that keeps integrating while a valve is fully open delays correction after disturbance reversal. 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 — Derivative action, filters, and noise. Use u_D=-Kd dy_f/dt to predict a measurable effect. Repeat tuning with the same process model but increasing noise to expose why derivative filtering matters. Explain how an ideal continuous derivative in a textbook is physically unrealizable in noisy instrumentation. 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 — Model-based tuning and robustness. Use Kc = tau/[K(lambda+theta)] to predict a measurable effect. Test tuning at ±20% gain, lag and delay variation and evaluate integral absolute error and valve travel. Explain how an apparently faster closed-loop design can fail when delayed measurement or valve saturation invalidates the nominal process model. 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 proportional offset; integral accumulation; derivative filtering; robust tuning and anti-windup. 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 4 assessment: PID Control and Tuning

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: Proportional action and steady offset

Question 1. Derive or interpret u=Kp(r-y). 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 (For plant static gain 2 and Kp=3, unit setpoint reference yields y_ss=6/7=0.857; the remaining error is 0.143 without integral action.), repeat the arithmetic, then alter one explicitly stated input by +15% and explain the sign of the difference.

Question 3. Challenge the approximation: The offset result assumes unity feedback, no bias term, a stable loop and no actuator saturation. Describe an experiment, instrument placement and observation that would invalidate this assumption.

Question 4. Troubleshoot the known risk: A noisy instrument can convert high proportional gain into rapid valve cycling and shortened actuator life. Propose a controlled test and a quantitative metric by which two possible causes can be separated.

Part 2: Integral action, bias and anti-windup

Question 5. Derive or interpret u=Kp e+Ki int(e dt). 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 (If Ki=0.20 actuator-units/(error-unit·s) and constant e=2 persists for 15 s, the integral contribution grows by 0.20215=6 actuator units before any limit is applied.), repeat the arithmetic, then alter one explicitly stated input by +15% and explain the sign of the difference.

Question 7. Challenge the approximation: A PI algorithm must specify integration units, discrete sample period, manual/auto transitions and saturation logic. Describe an experiment, instrument placement and observation that would invalidate this assumption.

Question 8. Troubleshoot the known risk: An integral term that keeps integrating while a valve is fully open delays correction after disturbance reversal. Propose a controlled test and a quantitative metric by which two possible causes can be separated.

Part 3: Derivative action, filters, and noise

Question 9. Derive or interpret u_D=-Kd dy_f/dt. 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 (A sensor fluctuation of 0.10 K over 0.02 s creates an unfiltered slope estimate of 5 K/s. With Kd=3 actuator-units·s/K, the instantaneous derivative contribution can reach 15 actuator units.), repeat the arithmetic, then alter one explicitly stated input by +15% and explain the sign of the difference.

Question 11. Challenge the approximation: Digital differentiation is a finite-difference approximation; the derivative filter adds lag and needs a documented bandwidth. Describe an experiment, instrument placement and observation that would invalidate this assumption.

Question 12. Troubleshoot the known risk: An ideal continuous derivative in a textbook is physically unrealizable in noisy instrumentation. Propose a controlled test and a quantitative metric by which two possible causes can be separated.

Part 4: Model-based tuning and robustness

Question 13. Derive or interpret Kc = tau/[K(lambda+theta)]. 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 (With K=2, tau=60 s, theta=10 s, and lambda=40 s, Kc=60/[2*(40+10)]=0.60; a common FOPDT PI rule sets Ti≈tau=60 s for this simplified case.), repeat the arithmetic, then alter one explicitly stated input by +15% and explain the sign of the difference.

Question 15. Challenge the approximation: This is one illustrative approximate tuning relation, not a universal optimum. Process type, objective and constraints matter. Describe an experiment, instrument placement and observation that would invalidate this assumption.

Question 16. Troubleshoot the known risk: An apparently faster closed-loop design can fail when delayed measurement or valve saturation invalidates the nominal process model. Propose a controlled test and a quantitative metric by which two possible causes can be separated.

Integrative design problem

For the case of heated-tank temperature regulation with actuator limits, 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.