PID Controllers

PID Controllers

• Proportional controllers operate correctly when deviations are not too large or sudden.
• The output is directly proportional to the error signal.
• A(t) ∝ e(t) represents the proportional controller output being directly proportional to the error signal.
• A(t) = Kp * e(t) is the equation when removing the sign of proportionality, where Kp is the proportional constant/controller gain.
• Kp should be greater than unity to amplify the error signal for easy detection.
• Proportional controllers reduce steady-state error and stabilize the system.
• Proportional controllers introduce offsets and increase the maximum overshoot.
• Proportional control action arrests changes and holds them steady, but at a different point from the desired value, with the difference being the "offset".
• Offset can be reduced by increasing system sensitivity via a narrower proportional band.
• Pneumatic proportional controllers use bellows to establish a set value and feed in the measured value.
• Differences in pressure cause movement, altering airflow and output pressure to the controlled valve.
• Moving the nozzle relative to the flapper increases sensitivity and gain, narrowing proportional bandwidth.
• The top end of the flapper moves oppositely to the lower end, reducing sensitivity and widening the proportional band.
• Objectives when adjusting a controller to a plant are minimum offset and stability.
• Proportional control provides throttling action to balance process input with demand.
• A set point is where it's desired to maintain the measured variable
• Offset is the deviation between the set point and the control point.
• Measurements stabilize within the proportional band, but narrowing it too much causes on-off controller cycling.

Integral Controllers

• Output is directly proportional to the integral of the error signal; also known as the actuating signal
• A(t) ∝ ∫e(t)dt is the equation for this relationship
• A(t) = Ki * ∫e(t)dt is the equation after removing the sign of proportionality, where Ki is the integral constant/controller gain.
• Integral controllers can be referred to as reset controllers.
• Reset controllers can return the controlled variable to the set point after a disturbance.
• A disadvantage is that it can make the system unstable by responding slowly to error.
• Proportional and integral action (P+I) can arrest change and reset to the desired value, load independent.
• Integral action happens continuously while deviation exists.
• Controllers have reset time adjustments, shorter times mean greater integral action, but too much integral effect causes overshoot.

Integral Corrective Action

• If an offset is acceptable, using a simple proportional controller may be best.
• Controllers can be adjusted to vary the integral action time or 'reset time'.
• Integral action is directly related to proportional control action.
• Integral action time is the time it takes for integral action to alter the controller output by an amount equal to the proportional action.
• As processes become harder to control, the proportional band must be increased to eliminate cycling.
• To maintain measurement, a control function can be introduced to maintain the measurement near a precise point, insteading of within the band

Derivative Controllers

• These are never used alone, but in combinations with other modes of controllers
• Derivative controllers cannot improve steady-state error, produce saturation effects, and amplify noise signals.
• The output is proportional to the derivative of the error signal.
• A(t) ∝ de(t)/dt is the equation for this relationship.
• A(t) = Kd * de(t)/dt is the equation after removing the sign of proportionality, where Kd is the proportional constant/controller gain.
• A derivative controller can be referred to as a rate controller.
• Derivative controllers improve the transient response of the system.
• The longer the derivative action time (adjustable), the greater the derivative action.
• Integral action is needed to eliminate offset with minimal overshoot.
• A third-term derivative control action can be a damping action to reduce overshoot.
• Derivative control action is anticipatory and deviations in the rate of change can estimate the final value of a measured variable
• A control signal based on the rate of change of a deviation can over-correct the controlled condition, reducing deviations.
• Rapid changes cause large overshoots, slow changes give small overshoots.
• Derivative control sets up a correction signal proportional to the variable's velocity when the variable is moving.
• Proportional mode produces a response proportional to deviation from the set point.
• Integral response is proportional to the time the measurement is away from the set point.
• Derivative response is proportional to the rate at which the measurement departs.
• A derivative control mode, when added to proportional-plus-integral-plus-derivative control, allows for a narrower proportional band reducing measurement deviation and overshoot.
• Pneumatic proportional-plus-integral-plus-derivative controllers differ from proportional-plus-integral ones with an adjustable derivative resistance (restrictor) in the feedback circuit.
• When the measurement is at the set point, the pressures in the proportional and integral bellows are equal to the output pressure of the controller.
• If the measurement increases uniformly, the derivative restrictor delays the output pressure from reaching the bellows, delaying the effect of reducing output, maintaining constant differential pressure.

PID Control Summary

• (P) indicates a Single term controller
• (P+I) or (P+D) indicates a Two term controller
• (P+I+D) indicates a Three term controller

PID Controller Tuning

• When manually setting a PID controller, first set the Ki (integral) and Kd (derivative) values to zero.
• Increase Kp (proportional) until the output of the loop oscillates, then set Kp to about half.
• Increase Ki until the offset is corrected in a timeframe, but be aware that too much Ki will cause instability.
• Increase Kd until the response is quick enough after a load change, but again be aware that too much results in excessive response and overshoot.
• Fast-acting PID loops usually overshoot to reach the setpoint quicker. But some systems cannot accept overshoot, where an over-damped closed loop system is needed.
• The Kp setting will need to be significantly less than half the initial value that was causing the oscillation.
• Manual calculation has been replaced with PID tuning and loop optimization software to ensure consistent results. The software will then gather data, develop process models and suggest optimal tuning.
• Some digital loop controllers feature a self-tuning process using small set point changes for optimal tuning values.

Limitations of PID Controllers

• PID controllers can perform poorly in some applications without optimal control.
• The fundamental issue is that it is a feedback control system, with constant parameters, where there is no knowledge of the process.
• For this reason performance is reactive and is a compromise
• PID control is the most effective when there is an observer capacity without a process model.
• Better performance can be attained by overtly modeling the process

PID Improvements

• A significant improvement is incorporating feed-forward control using knowledge about the system
• PID controllers can be changed using gain scheduling, adaptive modifications, improved measurement or cascading multiple PID controllers.
• Control system performance can be improved with feed-forward (open-loop) control, combined with feedback (closed-loop) PID control.
• Feed-forward values can provide most of the controller output.
• The PID controller has to compensate for any difference or error remains between the set point and system response to the open loop control.
• Since its unaffected by the process feedback, it can never cause the control system to oscillate, improving system response and stability.
• Feed-forward can be based on the set point and any extra measured disturbances.

Control System Loops: Split Range Control

• Split-range control uses one controller to manage two final-control elements

• Typically found in temperature control applications but extends far beyond this.

• In temperature control applications, the process could need heating or cooling dependent on product temperate

• The controller output range (0% – 100%) is split between two valves.

• If the controller output:

  • is between 0%–50%, then the cooling valve operates, fully open at 0% and fully closed at 50%
  • is between 50%–100%, then the heating valve operates, starts to open at 50% and is fully open at 100%

• In split-range control, there are different ways to connect the valves so they operate on two different ranges.

• Current-to-pressure converters split controller output; one responding to 4mA to 12mA, the other to 12mA to 20mA.

• Split-range can happen with pneumatic signals; the two valves are mechanically designed differently, using spring/diaphragm actuators.

• Cooling valve opens/closes from 3psi–8psi; the heating valve closes/opens from 8psi–13psi.

Dead Band for Split Range Control

• Continuous switching between cooling and heating near 50% output may arise
• A dead band avoids oscillation. Set the cooling valve range from 0%–49% and the heating valve range from 51%–100%.
• In engine thermal load, the slave controller detects lube oil temperature changes from the cooler, compares it to the set value and adjust is seawater flow through the cooler.
• With constant seawater temperature and an engine thermal-load drop, the master controller sense is a piston cooling oil temperature drop, and adjust is the seawater flow through the cooler.
• If the engine thermal load is low or zero, steam diaphragm valve (1) receives signal from slave controller, which supplies steam to the lube oil heater.
• Means that the slave control is split between valve positioners (1) and (2), which is called "split range control" or "split level control".
• The slave controller output range is between 1.2 and 2.0 bar. - Valve positioner (1) works on the range 1.2 to 1.4 bar. - Valve positioner (2) works on the range 1.4 to 2.0 bar.
• The range is split in the ratio 1:3.

Control System Loops - Ratio Control

• Ratio control maintains set ratios between two or more flow rates
• In a ratio-control scenario, one flow, "master flow", aligns with production rate.
• The ratio controller regulates another flow to maintain the desired ratio between the two flows, which is called the "controlled flow".
• In treating water with chlorine, water is the master flow while chlorine is the controlled flow.
• Ratio control should calculate a flow set point, not a division of flow rates.

Control System Loops - Cascade Control

• A product temperature regulated via heated oil jacket utilizes cascade control.
• The maximum input (400°C) restricts jacket temperature.
• At start-up, the master compares the product temperature (ambient) to its setpoint (300°C) to give maximum output.
• This sets the slave’s maximum setpoint (400°C); the slave compares it to the jacket temperature (ambient) giving maximum heater output.
• As the jacket temperature rises towards setpoint, the slave’s heater output decreases.
• The master's PID output decreases, reducing the ‘jacket’ setpoint on the slave, when product temperature rises.
• Balanced occurs when continuing this process
• Quicker, smoother control results while coping with load changes by keeping the jacket temperature within acceptable tolerances.

Control System Loops - Two Element Control

• Steam flow and water level measurements define two-element control.
• Sends out a pneumatic proportional to changes in its measured variable.
• The steam flow signal goes to a relay. This relay transmits signal to the valve positioner, via hand auto unit, which alters the valve position an amount proportional to the steam flow.
• Characteristics in the valve positioner are adjusted to match changes with feed
• The water level signal goes to a P+I controller whose set value adjusts to keep the boiler water at a desired level is set.
• Any deviation of the water level and the desired values results in a change in output signal from the controller to the relay.
• This signal is added to the steam flow signal to correct the deviation in level which could occur due to unbalance of boiler feed and steam flow.

Process Control

• A basic control example is keeping water temperature constant at cleaning stations despite demand.
• Manual control uses a thermometer and operator adjustment to maintain the temperature.
• Automated control senses the water temperature, sends an electrical signal to a controller, which sends a correcting signal to an actuator; the actuator adjusts steam flow to maintain the temperature.
• Process control automatically manages output by comparing a measured output to the desired "set value," adjusting an input variable based on the error signal.

Additional Process Control Info

• A process manipulates inputs to create a new output, and controlled inputs and measured output parameters make up variables.

• Rather than being limited to one variable, a controller can measure and control many variables, for example, as an engine processor does.

• Functions performed by an engine processor, typically over six to eight devices depending on the number of cylinders, have to be within approximately 5 milliseconds.

• Examples of measured and controlled variables in the Automotive Engine:

  • Measured Variables: Manifold Air Pressure, Manifold Air Flow, Oxygen in Exhaust
  • Controlled Variables: Ignition Timing, Injector Timing, Fuel Flow

• The measuring element comprises a sensor, transducer, and transmitter plus regulated power, while the control element comprises actuator, power control circuit and power. The controller comprises a processor with a memory and summing circuit to compare set point with sensed signal. The processor uses the error to generate a correction signal to control the actuator and the input variable.

Process Control Terminology

• Feedback loop: the signal path from output to input that corrects variations between output and setpoint where the output of a process is being continually monitored. The error is determined, and a correction signal is sent back to inputs.
• Controlled/measured variable: the monitored process output value.
• Manipulated variable: the input that a control signal varies via an actuator.
• Set point: the desired output value monitored by a sensor, where any deviation creates an error signal.
• Instrument: any device indicating/measuring physical quantities/conditions.
• Sensors: detect physical variables; examples include thermometers (temperature), or diaphragm pressure sensors with strain transducers.
• Transducers: change energy form (resistance thermometer converting temp to resistance).
• Converters: change signal format (voltage-to-current).
• Actuators: control an input variable in response to controller signals with a typical example being a flow-control valve.
• Controllers: Monitor signals from transducers to keep the process within limits.
• Programmable Logic Controllers (PLCs): Microprocessor-based systems used in process control. They use analog/digital I/O, communicate globally, are easily programmed, and supply extensive data.
• Ladder networks are typically used to program these controllers
• Error signal: the difference between the "set point" and the amplitude of the measured variable.
• Correction signal: controls actuator power to set the input level.
• Transmitters: amplify/format signals for long-distance transmission of data, or digital signals that can be transmitted without information loss.

Integrated Circuits

• Integrated circuit is a set of electronic circuits on a silicon piece, and can be called an IC, chip, or microchip

• Integrated circuits can be:

  • analog
  • digital
  • mixed signal with both analog and digital

• Digital integrated circuits squares millimeters and can contain billions of logic gates, multiplexers, flip-flops, etc.

• These digital integrated circuits require little, low power and speed to operate, used in microprocessors and microcontrollers to process ‘“one” and “zero” signals with Boolean algebra.

• Microprocessors, which control computers and cellular phones, are among the most advanced integrated circuits.

• Digital memory chips and application-specific integrated circuits (ASICs) are used in modern applications.

• Programmable logic devices came about in the 1980s

• Logical function and connectivity is able to be programmed by the user, rather than the manufacture, so a chip can implement different functions such as logic gates, adders, and registers.

• Analog ICs (sensors, power management, operational amplifiers), work in continuous signals.

• Mixed chips combine analog and digital circuits for A/D and D/A converters, while requiring concern for signal interference.

• Mono-crystalline silicon is the main substance used, with gallium arsenide for LEDs, lasers etc.

• A random-access memory (RAM) is a typical IC, and material layers are fabricated like a photographic.

Logic Gates

• Combining Gates enables complex operations, limited in practice
• Arrays of Logic Gates are found in Digital Integrated Circuits (ICs).
  • Digital Devices of the same size shrink as logic gate volume gets smaller and become more capable.

The AND Gate

• Act in the same way as the logical and operator
• the output is true when both inputs are true other wise it is false

The OR Gate

• Its output is true of either or both inputs are true otherwise it is false

Derived and Inverter Gates (XOR)

• The XOR (exclusive OR) Gate acts in the same way as the logical other/or.
  • Output is true if either, but not both, inputs are true
  • Output is false if both inputs are true or both imputs arwe false
• In other words, the output is 1 if the inputs are different otherwise they are the same

Inverter

• Only has one input and is somtimes called a nor gate
• Reverses logic gate
• The NAND Gate a negation and and gate
  • it acts as the AND logical gate with the negation process the output is false if both imputs are true otherwise the output is true.
  • There is no fimited ofr inouts that cab be used in a NAND function nor a funktional amount of inputs that can be used to NAND
• The NOR gate is a combination OR that is followed by an invertor if both inputis are false output is true if not output if false
• The XNOR (exclusive NOR) gate is a combination XOR gate followed by an invertor. Unlike OR/NOR adn AND/NAND Functions XOR always has 2 imputs