Instrument Technician - Multi Variable Control Notes

Multi Variable Control Strategies

Objective One: Advantages and Applications of Multi Variable Control

Multi variable control is a control system that uses input signals from two or more process variables to jointly affect the action of the control system. The goal is to minimize the interaction between two or more control variables.

Example: Level vessel process where level & outflow interact; outflow increase causes level drop.

Two multi variable control strategies:

  1. Centralized Control

  2. Decoupling Control.

Centralized Control

Uses all measurements to calculate all outputs simultaneously. It uses a process model to determine the outputs to the final control elements (FCEs).

  • Process models can be derived from fundamental principles or developed experimentally.

  • The controller is a computer program using the process model, control objectives, and measurement signals to calculate FCE outputs.

  • Examples: Model Predictive Control, Dynamic Matrix Control.

Decoupling Control

Compensates interacting control loops automatically when any single control loop takes a control action. Decoupling is used with multi-loop control strategies that interact. A multi-loop strategy uses multiple single-loop controllers to control the process.

  • Multi-loop strategy uses multiple single-loop controllers

Loop Interaction

A multi variable process has loop interaction when a manipulated variable affects more than one controlled variable.

  • A change in the output of LC-100 (level control) changes the inflow, affecting the level but not the outflow (no interaction).

  • A change in the output of FC-100 (flow control) changes the outflow, affecting both the level and the outflow (interaction).

Loop interaction can significantly impact process stability, potentially leading to unstable systems, such as cyclic responses in temperature control processes (e.g., hot and cold water mixing).

  • Multi-Loop control works well if interaction is minor.

  • Multi Variable Control or redesign is required for significant interaction.

A multi variable control strategy reacts to changes by adjusting multiple inputs simultaneously to prevent loop interaction.

To determine the severity of loop interactions and the best pairing of controlled and manipulated variables, a relative gain matrix (RGM) development is necessary.

  • RGM can be developed from fundamental principles or determined experimentally.

  • OLM contains the static gains of each MV on each CV.

  • Each variable (MV or CV) has its own static gain.

To develop an Open Loop Matrix (OLM) experimentally for a 2 × 2 system:

  1. Put both controllers on manual and adjust their outputs to bring the controlled variables close to their normal setpoints.

  2. Wait until the system reaches steady state.

  3. Step the output of controller one (\Delta CO_1) and wait for the system to again reach steady state.

  4. Calculate the static gain for both process variables due to the step change in controller one.

    K{11} = \frac{\Delta PV1}{\Delta CO_1}

    K{21} = \frac{\Delta PV2}{\Delta CO_1}

  5. Step the output of controller one back to its original value and wait for the system to reach steady state.

  6. Step the output of controller two (\Delta CO_2) and wait for the system to again reach steady state.

  7. Calculate the static gain for both process variables due to the step change in controller two.

    K{12} = \frac{\Delta PV1}{\Delta CO_2}

    K{22} = \frac{\Delta PV2}{\Delta CO_2}

  • OLM is made with static gains (K).

  • RGM is calculated with the OLM.

The OLM can be developed by arranging the static gains as shown below.

OLM = \begin{bmatrix} K{11} & K{12} \ K{21} & K{22} \end{bmatrix}

The RGM is calculated from the OLM as follows:

RGM = \begin{bmatrix} \mu{11} & \mu{12} \ \mu{21} & \mu{22} \end{bmatrix}

Where:

\mu{11} = \mu{22} = \frac{K{11}K{22}}{K{11}K{22} - K{12}K{21}}

\mu{12} = \mu{21} = 1 - \frac{K{11}K{22}}{K{11}K{22} - K{12}K{21}}

  • u{11} & u{22} are Relative Gains.

*CO and PV must be paired to make relative gain as close to 1 to minimize effect of interaction.

*Relative Gain (u) of 0.69 means the loop gain increases by a factor of 1/0.69 = 1.5 when the other loop is changed from manual to auto.

*Negative relative gain indicates the process action switches when the other loop changes to automatic. 1 / 2.6 = 0.38 is the Loop Gain Change when other loop changes to automatic

***Relative gain pairings between 0.9 and 1.1 (less than 15% interaction) can be controlled without a multivariable control strategy. For values outside this range, the controllers must either be detuned or a multivariable control strategy must be used.

Decoupling Control Strategy

A decoupling control strategy is a multivariable strategy that minimizes the interaction between loops. In this control strategy, the output of each controller goes to both control valves.

  • The signal going to FV-100 is equal to CO2 + (-0.5 \times CO1).

  • The signal going to AC-100 is equal to CO1 + (0.88 \times CO2).

  • An output change from either controller goes to both valves.

*The Decoupler allows change of one output (i.e., composition) without a change of the other output (i.e., flow rate), and vice versa.

Objective Two: Block Diagrams of Multi Variable Control Systems

The relative gain matrix (RGM) provides a quantitative measure of loop interaction and can identify the best pairings for the controlled variable and the manipulated variable. In general, relative gain pairings between 0.90 and 1.10 can be controlled without a multivariable control strategy. Pairings outside this range have significant interaction, which a decoupling control strategy can reduce.

A decoupling control strategy uses decouplers to cancel the effects of the interaction between control loops. The design of the decouplers can be developed from a block diagram of the process that illustrates their implementation.

*Block Diagram shows how the process operates using transfer functions.
*Transfer Functions (G) are mathematical algorithms that describe how different parts of a control loop function.

The combined transfer functions are indicated by the subscript p. The second numerical subscript indicates which controller output is the transfer function input. The first subscript indicates the transfer function process variable output.

  • G{p11} describes how a change in CO1 affects the PV_1.

  • G{p21} describes how a change in CO1 affects the PV_2.

  • G{p22} describes how a change in CO2 affects the PV_2.

  • G{p12} describes how a change in CO2 affects the PV_1.

Decoupler Design

A decoupling control strategy can be designed from a block diagram of the process. This design uses two decouplers, D1 and D2. The purpose of the decouplers is to cancel the effects of loop interaction so that each control loop is not affected by actions from the other control loop. The purpose of D2 is to compensate for the effect that a change in CO2 has on PV_1.

To cancel the effect CO2 has on PV1, decoupler D_2 must have the following transfer function:

D2 = -\frac{G{p12}}{G_{p11}}

Similarly, the purpose of D1 is to compensate for the effect that a change in CO1 has on PV2. Following the signal path shows how a change in CO1 affects PV2 with CO2 constant.

(ACO1)(D1)(G{p22})+(ACO1)(G{p21}) = \Delta PV2

To cancel the effect CO1 has on PV2, the decoupler D_1 must have the following transfer function.

D1 = -\frac{G{p21}}{G_{p22}}

Static Decoupler

A static decoupler only uses the static gains of the transfer functions in its design. The decoupler designs are then as follows.

D1 = -\frac{K{21}}{K_{22}}

D2 = -\frac{K{12}}{K_{11}}

A static decoupler works best when the process dynamics of the two loops are similar. If the dynamics are dissimilar, a dynamic decoupler design can give better performance.

Dynamic Decoupler

A dynamic decoupler uses the static gains and dynamic components of the transfer functions in its design. The decoupler designs are as follows.

D1 = -\frac{G{21}}{G_{22}}

D2 = -\frac{G{12}}{G_{11}}

*Static Decouplers work best when the process dynamics of both loops are similar
*Proper transposition.

*Deadtime is NOT used when it has a positive value.
*Above is modeled after FOPDT transfer function.

Decouplers can be designed for integrating systems also.

Decoupler Design Example

The loop transfer functions G{11}, G{12}, G{21} and G{22} need to:

  • Confirm that the controller pairing is correct,

  • Determine the severity of the loop interaction and

  • Determine design appropriate decouplers.

*To determine G (transfer function), both loops must be in manual, and their PVs close to normal operating value. Then make a step change in CO1 while keeping CO2 constant. Then calculate Gain, Dead Time and FOTC (first order time constant).

*Same as above with opposite CO changes.

To determine whether the controller pairing is correct, and the severity of the loop interaction, you must first calculate the OLM and then the RGM.

Tuning without Decoupling

*Transfer Functions help determine relative speed of loops
*RGM helps determine the best pair looping and how the loops interact.
*Loop gain decreases when the other loops are in automatic (closed) to avoid becoming unstable.
*Loops must be tuned with the other loops in manual (open).

*Tune faster loops first with all other loops in manual
*Tune slower loops next from least to most important
*Last (most important) loop does NOT need to be tuned with all other loops in automatic
*Fine-tune testing must be done with other loops in manual and in automatic
*For any loop with RG < 1, reduce the controller's Proportional Gain by multiplying by its RG

*Same as table above, with
Fast Loop = Least Important Loop
Slow Loop = Most
*Tune faster loops (with slower loops in Manual)
Then lower proportional gain by multiplying by RG
Then slower loops (with faster loop in Automatic)
*For loops with similar speeds:
*Tune least important loop first with most important in Manual
Then lower proportional gain by multiplying by RG
*Tune most important loop with least important in Automatic

Tuning with Decoupling

*Decoupling does not change the effect on the control loop transfer function when the other loop is switched between manual and automatic.

Using the open loop transfer functions, the decouplers are designed as follows.

Configuring a Decoupling Strategy

*Decoupling control uses two SUM blocks to send the output of a controller to both control valves.

Control System Considerations

*Bumpless Transfer achieved by adding SP Tracking and External Feedback
*Prevent Reset Windup with External Feedback and Anti-Reset Windup
*Decoupler On/Off function

Nonlinear Processes

*Nonlinear Processes result in model errors (incorrect decoupling parameters)
*Large RG pairings in nonlinear processes create instability and may require further detuning

Partial Decoupling

*Partial Decoupling uses only one decoupler applied to the most important CV

Partial decoupling should be considered for interacting process where one controlled variable is more important than the other controlled variable to provide more stable control. In partial decoupling, only one decoupler is implemented.

Self-Test and Answers

(Refer to the original document for the self-test questions and answers.)