Advanced and Multivariable Control Strategies

Scope: cascade; feedforward; ratio/override; multivariable interaction. Cross-cutting case: heat-transfer train and blending manifold with coupled loops.

What this unit builds

A time response must be read together with the input that created it. A step response without the actuator's recorded movement can conceal valve saturation. A smooth trace without sensor location can conceal delay. When two mechanisms both explain the nominal trend, they must be tested using a disturbance or experiment that separates their effects, not selected on visual appearance alone.

Lesson sequence

1. [Cascade control for measurable secondary disturbances](../lessons/unit-06-lesson-01.html) — Cascade control uses a fast secondary measurement to reject disturbances before they propagate to a slower primary variable. The inner loop must be considerably faster, correctly paired and reliably measured. Governing relation: `u_outer=r_inner; u_inner=C_inner(r_inner-y_inner)`.

2. [Feedforward and measured disturbances](../lessons/unit-06-lesson-02.html) — Feedforward computes corrective action from a measured load disturbance before the controlled variable responds. It only works as predicted when the disturbance sensor, process model and actuator authority are adequate. Governing relation: `u_ff(s)= -Gd(s)/Gp(s) d(s)`.

3. [Ratio control, selectors, and constraints](../lessons/unit-06-lesson-03.html) — Ratio control coordinates one stream with another for blending or combustion-like material balance. Override selectors choose a constrained command based on safety or equipment limits, but selection and anti-windup must be explicit. Governing relation: `F_B=r F_A`.

4. [Multivariable interaction and decoupling](../lessons/unit-06-lesson-04.html) — Two manipulated variables can affect two controlled outputs; independent single-loop tuning may lead to competition. Relative gain and gain-matrix conditioning indicate possible pairing difficulties. Governing relation: `Delta y=G(0) Delta u`.

Model connections and physical interpretation

Cascade control for measurable secondary disturbances

Cascade control uses a fast secondary measurement to reject disturbances before they propagate to a slower primary variable. The inner loop must be considerably faster, correctly paired and reliably measured. A nested architecture requires anti-windup and mode coordination when the inner loop is manual or unavailable. The calculation that anchors this lesson is deliberately synthetic: A jacket inlet flow loop settles in 4 s while reactor temperature settles in 60 s; the approximate 15:1 separation makes a cascade architecture plausible for rejecting jacket flow disturbances. 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.

Feedforward and measured disturbances

Feedforward computes corrective action from a measured load disturbance before the controlled variable responds. It only works as predicted when the disturbance sensor, process model and actuator authority are adequate. Dynamic mismatch and sensor delay leave residual error; feedback should remain available to correct unmodeled effects. The calculation that anchors this lesson is deliberately synthetic: If steady process gain from valve is Gp=2 K/% and disturbance gain is Gd=−3 K/(kg/s), a +1 kg/s disturbance calls for u_ff=+1.5% in the ideal static cancellation. 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.

Ratio control, selectors, and constraints

Ratio control coordinates one stream with another for blending or combustion-like material balance. Override selectors choose a constrained command based on safety or equipment limits, but selection and anti-windup must be explicit. Fluid composition and density determine whether the ratio should be mass-, molar- or volume-based. The calculation that anchors this lesson is deliberately synthetic: At F_A=12 kg/min and target ratio r=0.25 kg B/kg A, required secondary flow is 3.0 kg/min. A 20% rise in primary flow should call for 3.6 kg/min if capacity permits. 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.

Multivariable interaction and decoupling

Two manipulated variables can affect two controlled outputs; independent single-loop tuning may lead to competition. Relative gain and gain-matrix conditioning indicate possible pairing difficulties. Dynamic interactions and actuator constraints can defeat a static inverse controller. The calculation that anchors this lesson is deliberately synthetic: For static gain matrix [[2,1],[1,2]], choose u=[1,0] to obtain outputs [2,1]; both variables respond, demonstrating interaction. Determinant=3, so the matrix is invertible in this ideal example. 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 heat-transfer train and blending manifold with coupled loops. 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 Cascade control for measurable secondary disturbances, state the terms of u_outer=r_inner; u_inner=C_inner(r_inner-y_inner), choose a limiting or equilibrium case, and explain when it would be invalid.
2. For Feedforward and measured disturbances, state the terms of u_ff(s)= -Gd(s)/Gp(s) d(s), choose a limiting or equilibrium case, and explain when it would be invalid.
3. For Ratio control, selectors, and constraints, state the terms of F_B=r F_A, choose a limiting or equilibrium case, and explain when it would be invalid.
4. For Multivariable interaction and decoupling, state the terms of Delta y=G(0) Delta u, choose a limiting or equilibrium case, and explain when it would be invalid.

Continue to the [Unit 6 assessment](../assessments/unit-06.html) or [Unit 6 project](../projects/unit-06.html).

Unit 6, Lesson 1: Cascade control for measurable secondary disturbances

Unit: Advanced and Multivariable Strategies. Physical setting: heat-transfer train and blending manifold with coupled loops. Reference model: u_outer=r_inner; u_inner=C_inner(r_inner-y_inner).

Learning outcomes

By the end of this lesson you should be able to (i) explain cascade control for measurable secondary disturbances in physical terms, (ii) identify the variables and assumptions of u_outer=r_inner; u_inner=C_inner(r_inner-y_inner), (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

Cascade control uses a fast secondary measurement to reject disturbances before they propagate to a slower primary variable. The inner loop must be considerably faster, correctly paired and reliably measured.

Consider the unit-wide equipment case: heat-transfer train and blending manifold with coupled loops. 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 cascade control for measurable secondary disturbances, where confusing a state with a measured output can give a mathematically solvable but physically incorrect model.

The chosen governing expression is u_outer=r_inner; u_inner=C_inner(r_inner-y_inner). 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 jacket inlet flow loop settles in 4 s while reactor temperature settles in 60 s; the approximate 15:1 separation makes a cascade architecture plausible for rejecting jacket flow disturbances.

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 cascade control for measurable secondary disturbances 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 nested architecture requires anti-windup and mode coordination when the inner loop is manual or unavailable. 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 cascade control for measurable secondary disturbances, 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 Cascade control for measurable secondary disturbances as a source of prior knowledge and the next topic Feedforward and measured disturbances as an independent challenge. Define the reference operating point for cascade control for measurable secondary disturbances 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 slow secondary loop can add lag and worsen primary regulation rather than improving it. 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: Apply the same jacket-supply disturbance to single-loop and cascade models and compare primary error and valve movement. 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 Ratio control, selectors, and constraints, 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 cascade control for measurable secondary disturbances, 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 11](../labs/lab-11.html) is Cascade control with a fast inner loop. 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_outer=r_inner; u_inner=C_inner(r_inner-y_inner), 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 cascade control for measurable secondary disturbances model appear wrong. What extra logged variable would distinguish that failure?
4. Explain how Ratio control, selectors, and constraints 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_outer=r_inner; u_inner=C_inner(r_inner-y_inner) 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 Advanced and Multivariable Strategies. 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 feedforward and measured disturbances is included in the model.

Unit 6, Lesson 2: Feedforward and measured disturbances

Unit: Advanced and Multivariable Strategies. Physical setting: heat-transfer train and blending manifold with coupled loops. Reference model: u_ff(s)= -Gd(s)/Gp(s) d(s).

Learning outcomes

By the end of this lesson you should be able to (i) explain feedforward and measured disturbances in physical terms, (ii) identify the variables and assumptions of u_ff(s)= -Gd(s)/Gp(s) d(s), (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

Feedforward computes corrective action from a measured load disturbance before the controlled variable responds. It only works as predicted when the disturbance sensor, process model and actuator authority are adequate.

Consider the unit-wide equipment case: heat-transfer train and blending manifold with coupled loops. 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 feedforward and measured disturbances, where confusing a state with a measured output can give a mathematically solvable but physically incorrect model.

The chosen governing expression is u_ff(s)= -Gd(s)/Gp(s) d(s). 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 steady process gain from valve is Gp=2 K/% and disturbance gain is Gd=−3 K/(kg/s), a +1 kg/s disturbance calls for u_ff=+1.5% in the ideal static cancellation.

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 feedforward and measured disturbances 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

Dynamic mismatch and sensor delay leave residual error; feedback should remain available to correct unmodeled effects. 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 feedforward and measured disturbances, 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 Cascade control for measurable secondary disturbances as a source of prior knowledge and the next topic Ratio control, selectors, and constraints as an independent challenge. Define the reference operating point for feedforward and measured disturbances 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 incorrect sign in feedforward can amplify the very upset it was intended to suppress. 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 cancellation with perfect, delayed and 25%-biased disturbance measurements. 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 Multivariable interaction and decoupling, 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 feedforward and measured disturbances, 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 12](../labs/lab-12.html) is Feedforward disturbance compensation. 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_ff(s)= -Gd(s)/Gp(s) d(s), 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 feedforward and measured disturbances model appear wrong. What extra logged variable would distinguish that failure?
4. Explain how Multivariable interaction and decoupling 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_ff(s)= -Gd(s)/Gp(s) d(s) 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 Advanced and Multivariable Strategies. 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 ratio control, selectors, and constraints is included in the model.

Unit 6, Lesson 3: Ratio control, selectors, and constraints

Unit: Advanced and Multivariable Strategies. Physical setting: heat-transfer train and blending manifold with coupled loops. Reference model: F_B=r F_A.

Learning outcomes

By the end of this lesson you should be able to (i) explain ratio control, selectors, and constraints in physical terms, (ii) identify the variables and assumptions of F_B=r F_A, (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

Ratio control coordinates one stream with another for blending or combustion-like material balance. Override selectors choose a constrained command based on safety or equipment limits, but selection and anti-windup must be explicit.

Consider the unit-wide equipment case: heat-transfer train and blending manifold with coupled loops. 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 ratio control, selectors, and constraints, where confusing a state with a measured output can give a mathematically solvable but physically incorrect model.

The chosen governing expression is F_B=r F_A. 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. At F_A=12 kg/min and target ratio r=0.25 kg B/kg A, required secondary flow is 3.0 kg/min. A 20% rise in primary flow should call for 3.6 kg/min if capacity permits.

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 ratio control, selectors, and constraints 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

Fluid composition and density determine whether the ratio should be mass-, molar- or volume-based. 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 ratio control, selectors, and constraints, 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 Feedforward and measured disturbances as a source of prior knowledge and the next topic Multivariable interaction and decoupling as an independent challenge. Define the reference operating point for ratio control, selectors, and constraints 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 ratio controller cannot enforce composition if feed composition varies unknown to the system. 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: Explore ratio error under flowmeter bias and a secondary valve reaching capacity. 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 Cascade control for measurable secondary disturbances, 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 ratio control, selectors, and constraints, 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 17](../labs/lab-17.html) is Ratio-controlled blending. 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_B=r F_A, 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 ratio control, selectors, and constraints model appear wrong. What extra logged variable would distinguish that failure?
4. Explain how Cascade control for measurable secondary disturbances 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_B=r F_A 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 Advanced and Multivariable Strategies. 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 multivariable interaction and decoupling is included in the model.

Unit 6, Lesson 4: Multivariable interaction and decoupling

Unit: Advanced and Multivariable Strategies. Physical setting: heat-transfer train and blending manifold with coupled loops. Reference model: Delta y=G(0) Delta u.

Learning outcomes

By the end of this lesson you should be able to (i) explain multivariable interaction and decoupling in physical terms, (ii) identify the variables and assumptions of Delta y=G(0) Delta u, (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

Two manipulated variables can affect two controlled outputs; independent single-loop tuning may lead to competition. Relative gain and gain-matrix conditioning indicate possible pairing difficulties.

Consider the unit-wide equipment case: heat-transfer train and blending manifold with coupled loops. 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 multivariable interaction and decoupling, where confusing a state with a measured output can give a mathematically solvable but physically incorrect model.

The chosen governing expression is Delta y=G(0) Delta u. 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 static gain matrix [[2,1],[1,2]], choose u=[1,0] to obtain outputs [2,1]; both variables respond, demonstrating interaction. Determinant=3, so the matrix is invertible in this ideal example.

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 multivariable interaction and decoupling 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

Dynamic interactions and actuator constraints can defeat a static inverse controller. 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 multivariable interaction and decoupling, 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 Ratio control, selectors, and constraints as a source of prior knowledge and the next topic Multivariable interaction and decoupling as an independent challenge. Define the reference operating point for multivariable interaction and decoupling 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: Pairing loops using only largest static gain may ignore delay and unfavorable high-frequency interaction. 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 one actuator at a time and identify all output effects before choosing diagonal loops or decoupling. 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 Feedforward and measured disturbances, 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 multivariable interaction and decoupling, 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 13](../labs/lab-13.html) is Two-tank interaction model. 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 Delta y=G(0) Delta u, 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 multivariable interaction and decoupling model appear wrong. What extra logged variable would distinguish that failure?
4. Explain how Feedforward and measured disturbances might confound the identification of this lesson’s mechanism and propose a discriminating test.
5. State one safety or operating limit that the simplified Delta y=G(0) Delta u 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 Advanced and Multivariable Strategies. 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 multivariable interaction and decoupling is included in the model.

Unit 6 applied project: Advanced and Multivariable Strategies

Decision problem

You are an analyst assigned to heat-transfer train and blending manifold with coupled loops. 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 — Cascade control for measurable secondary disturbances. Use u_outer=r_inner; u_inner=C_inner(r_inner-y_inner) to predict a measurable effect. Apply the same jacket-supply disturbance to single-loop and cascade models and compare primary error and valve movement. Explain how a slow secondary loop can add lag and worsen primary regulation rather than improving it. 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 — Feedforward and measured disturbances. Use u_ff(s)= -Gd(s)/Gp(s) d(s) to predict a measurable effect. Test cancellation with perfect, delayed and 25%-biased disturbance measurements. Explain how an incorrect sign in feedforward can amplify the very upset it was intended to suppress. 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 — Ratio control, selectors, and constraints. Use F_B=r F_A to predict a measurable effect. Explore ratio error under flowmeter bias and a secondary valve reaching capacity. Explain how a ratio controller cannot enforce composition if feed composition varies unknown to the system. 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 — Multivariable interaction and decoupling. Use Delta y=G(0) Delta u to predict a measurable effect. Vary one actuator at a time and identify all output effects before choosing diagonal loops or decoupling. Explain how pairing loops using only largest static gain may ignore delay and unfavorable high-frequency interaction. 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 cascade; feedforward; ratio/override; multivariable interaction. 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 6 assessment: Advanced and Multivariable Strategies

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: Cascade control for measurable secondary disturbances

Question 1. Derive or interpret u_outer=r_inner; u_inner=C_inner(r_inner-y_inner). 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 jacket inlet flow loop settles in 4 s while reactor temperature settles in 60 s; the approximate 15:1 separation makes a cascade architecture plausible for rejecting jacket flow disturbances.), repeat the arithmetic, then alter one explicitly stated input by +15% and explain the sign of the difference.

Question 3. Challenge the approximation: A nested architecture requires anti-windup and mode coordination when the inner loop is manual or unavailable. Describe an experiment, instrument placement and observation that would invalidate this assumption.

Question 4. Troubleshoot the known risk: A slow secondary loop can add lag and worsen primary regulation rather than improving it. Propose a controlled test and a quantitative metric by which two possible causes can be separated.

Part 2: Feedforward and measured disturbances

Question 5. Derive or interpret u_ff(s)= -Gd(s)/Gp(s) d(s). 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 steady process gain from valve is Gp=2 K/% and disturbance gain is Gd=−3 K/(kg/s), a +1 kg/s disturbance calls for u_ff=+1.5% in the ideal static cancellation.), repeat the arithmetic, then alter one explicitly stated input by +15% and explain the sign of the difference.

Question 7. Challenge the approximation: Dynamic mismatch and sensor delay leave residual error; feedback should remain available to correct unmodeled effects. Describe an experiment, instrument placement and observation that would invalidate this assumption.

Question 8. Troubleshoot the known risk: An incorrect sign in feedforward can amplify the very upset it was intended to suppress. Propose a controlled test and a quantitative metric by which two possible causes can be separated.

Part 3: Ratio control, selectors, and constraints

Question 9. Derive or interpret F_B=r F_A. 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 (At F_A=12 kg/min and target ratio r=0.25 kg B/kg A, required secondary flow is 3.0 kg/min. A 20% rise in primary flow should call for 3.6 kg/min if capacity permits.), repeat the arithmetic, then alter one explicitly stated input by +15% and explain the sign of the difference.

Question 11. Challenge the approximation: Fluid composition and density determine whether the ratio should be mass-, molar- or volume-based. Describe an experiment, instrument placement and observation that would invalidate this assumption.

Question 12. Troubleshoot the known risk: A ratio controller cannot enforce composition if feed composition varies unknown to the system. Propose a controlled test and a quantitative metric by which two possible causes can be separated.

Part 4: Multivariable interaction and decoupling

Question 13. Derive or interpret Delta y=G(0) Delta u. 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 (For static gain matrix [[2,1],[1,2]], choose u=[1,0] to obtain outputs [2,1]; both variables respond, demonstrating interaction. Determinant=3, so the matrix is invertible in this ideal example.), repeat the arithmetic, then alter one explicitly stated input by +15% and explain the sign of the difference.

Question 15. Challenge the approximation: Dynamic interactions and actuator constraints can defeat a static inverse controller. Describe an experiment, instrument placement and observation that would invalidate this assumption.

Question 16. Troubleshoot the known risk: Pairing loops using only largest static gain may ignore delay and unfavorable high-frequency interaction. Propose a controlled test and a quantitative metric by which two possible causes can be separated.

Integrative design problem

For the case of heat-transfer train and blending manifold with coupled loops, 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.