3.4 PID controllers

PID controllers are the workhorses of industrial control systems. They use proportional, integral, and derivative actions to maintain desired setpoints by continuously calculating and correcting errors between setpoints and measured variables.

PID controllers operate in closed-loop feedback systems, adjusting outputs based on error signals. They balance fast response with stability, eliminating steady-state errors while minimizing overshoot and oscillations. Proper tuning is crucial for optimal performance.

PID controller overview

• PID controllers are widely used in industrial control systems to regulate process variables and maintain desired setpoints
• Combines proportional, integral, and derivative control actions to achieve robust and efficient control performance
• Continuously calculates an error value as the difference between a desired setpoint and a measured process variable and applies a correction based on proportional, integral, and derivative terms

Proportional, integral, derivative terms

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• Proportional term produces an output value that is proportional to the current error value
  • Larger the error, greater the proportional control action
• Integral term considers the past values of the error and integrates them over time to eliminate steady-state error
  • Accumulates the error over time and provides necessary action to eliminate the residual steady-state error
• Derivative term predicts the future behavior of the error based on its current rate of change
  • Improves settling time and stability of the system by reducing overshoot and oscillations

Feedback control loop

• PID controller operates in a closed-loop feedback control system
• Continuously measures the process variable using sensors or transmitters
• Compares the measured value with the desired setpoint to calculate the error signal
• Adjusts the control output based on the calculated PID control action to minimize the error and maintain the process variable at the setpoint

Error signal calculation

• Error signal represents the difference between the desired setpoint and the measured process variable
• Calculated in real-time by subtracting the measured value from the setpoint
  • Positive error indicates the process variable is below the setpoint
  • Negative error indicates the process variable is above the setpoint
• PID controller uses the error signal as input to determine the necessary control action

Proportional control

• Proportional control action is directly proportional to the error signal
• Adjusts the control output based on the magnitude of the error
• Provides a rapid response to changes in the process variable and helps reduce the error quickly

Proportional gain

• Proportional gain ($K_p$) determines the strength of the proportional control action
• Higher proportional gain results in larger changes in the control output for a given error
• Affects the system's responsiveness, rise time, and stability
  • Too high $K_p$ can lead to oscillations and instability
  • Too low $K_p$ can result in sluggish response and poor control performance

Steady-state error

• Proportional control alone often results in a steady-state error (offset) between the setpoint and the actual process variable
• Occurs because the proportional action diminishes as the error approaches zero
• Steady-state error can be reduced by increasing the proportional gain, but it cannot be eliminated completely without introducing other control actions (integral or derivative)

Rise time vs overshoot

• Proportional gain affects the rise time and overshoot of the system response
• Higher proportional gain reduces the rise time, allowing the process variable to reach the setpoint faster
• However, high proportional gain can also lead to overshoot, where the process variable exceeds the setpoint before settling
• Trade-off between fast response (short rise time) and stability (low overshoot) must be considered when tuning the proportional gain

Integral control

• Integral control action considers the accumulated error over time
• Eliminates the steady-state error that occurs with proportional control alone
• Continuously integrates the error signal and adjusts the control output accordingly

Integral gain

• Integral gain ($K_i$) determines the strength of the integral control action
• Higher integral gain results in faster elimination of steady-state error but can lead to oscillations and instability if set too high
• Integral gain is typically set lower than the proportional gain to maintain stability while still providing the necessary integral action

Eliminating steady-state error

• Integral control action continues to accumulate the error over time, even when the error is small
• Drives the control output in the direction that reduces the error, eventually eliminating the steady-state error
• Ensures that the process variable accurately tracks the setpoint in the steady-state condition

Windup phenomenon

• Integral windup occurs when the integral term accumulates a significant error during periods of large setpoint changes or system saturation
• Leads to overshoots, long settling times, and potential instability
• Anti-windup techniques are employed to prevent integral windup
  • Clamping the integral term within predefined limits
  • Conditional integration (stopping integration when certain conditions are met)
  • Back-calculation and tracking

Derivative control

• Derivative control action responds to the rate of change of the error signal
• Anticipates the future behavior of the error and provides a corrective action before the error becomes too large
• Improves the system's stability and transient response by reducing overshoot and oscillations

Derivative gain

• Derivative gain ($K_d$) determines the strength of the derivative control action
• Higher derivative gain results in more aggressive correction based on the rate of change of the error
• Derivative gain is typically set lower than the proportional and integral gains to avoid excessive sensitivity to noise and high-frequency disturbances

Improving transient response

• Derivative control action helps improve the transient response of the system
• Reduces overshoot by counteracting the rapid changes in the error signal
• Provides a damping effect, helping the system settle faster and reducing oscillations
• Particularly effective in systems with significant time delays or inertia

Noise amplification

• Derivative control action is sensitive to noise and high-frequency disturbances in the measured process variable
• Amplifies the noise, leading to erratic control behavior and potential instability
• Filtering techniques, such as low-pass filters or signal smoothing, are often employed to mitigate the effects of noise on the derivative term
• Careful tuning of the derivative gain is necessary to balance the benefits of improved transient response with the sensitivity to noise

PID tuning methods

• PID tuning involves selecting appropriate values for the proportional, integral, and derivative gains to achieve the desired control performance
• Various tuning methods have been developed to systematically determine the optimal PID parameters based on the system characteristics and performance requirements

Manual tuning

• Manual tuning relies on trial-and-error adjustments of the PID gains based on the observed system response
• Involves gradually increasing the proportional gain until the system oscillates, then adjusting the integral and derivative gains to achieve the desired response
• Requires experience and intuition to achieve satisfactory results
• Suitable for simple systems or when a rough initial tuning is sufficient

Ziegler-Nichols method

• Ziegler-Nichols method is a widely used empirical tuning approach
• Based on the system's response to a step input or the ultimate gain and period of sustained oscillations
• Provides a set of tuning rules to determine the PID gains based on the critical gain ($K_u$) and critical period ($T_u$) of the system
  • Proportional gain: $K_p = 0.6K_u$
  • Integral time: $T_i = 0.5T_u$
  • Derivative time: $T_d = 0.125T_u$
• Offers a good starting point for further fine-tuning

Cohen-Coon method

• Cohen-Coon method is another empirical tuning approach based on process reaction curve
• Considers the process gain, dead time, and time constant to determine the PID gains
• Provides a set of equations to calculate the PID parameters based on the process characteristics
• Suitable for systems with a first-order plus dead time (FOPDT) model
• Tends to produce more conservative tuning compared to the Ziegler-Nichols method

PID controller design

• PID controller design involves selecting the appropriate structure and parameters to meet the control objectives and system requirements
• Different forms of PID controllers are used depending on the application and implementation constraints

Continuous-time PID

• Continuous-time PID controller operates in the time domain
• Described by the parallel or standard form of the PID algorithm
  • Parallel form: $u(t) = K_p e(t) + K_i \int e(t)dt + K_d \frac{de(t)}{dt}$
  • Standard form: $u(t) = K_p (e(t) + \frac{1}{T_i} \int e(t)dt + T_d \frac{de(t)}{dt})$
• Assumes continuous measurement and control action
• Suitable for analog implementations or systems with fast sampling rates

Discrete-time PID

• Discrete-time PID controller operates in the sampled-data domain
• Approximates the continuous-time PID algorithm using numerical integration and differentiation methods
• Described by the velocity or positional form of the discrete PID algorithm
  • Velocity form: $\Delta u(k) = K_p (e(k) - e(k-1)) + K_i T_s e(k) + \frac{K_d}{T_s} (e(k) - 2e(k-1) + e(k-2))$
  • Positional form: $u(k) = u(k-1) + K_p (e(k) - e(k-1)) + K_i T_s e(k) + \frac{K_d}{T_s} (e(k) - 2e(k-1) + e(k-2))$
• Suitable for digital implementations and computer-based control systems

Setpoint weighting

• Setpoint weighting is a technique used to modify the PID algorithm to improve the system response to setpoint changes
• Introduces separate weights for the proportional ($b$) and derivative ($c$) actions on the setpoint
  • Modified PID algorithm: $u(t) = K_p (b r(t) - y(t)) + K_i \int (r(t) - y(t))dt + K_d (\frac{d}{dt}(c r(t) - y(t)))$
• Allows independent tuning of the system's response to setpoint changes and disturbance rejection
• Helps reduce overshoot and improve the overall control performance

PID controller limitations

• PID controllers, despite their widespread use, have certain limitations that need to be considered when applying them to real-world systems
• Understanding these limitations helps in determining the suitability of PID control for a given application and guides the selection of alternative control strategies when necessary

Nonlinear systems

• PID controllers are designed based on linear control theory and assume a linear relationship between the input and output of the system
• Nonlinear systems exhibit complex behaviors and may have varying gains, dead zones, or saturation limits
• PID controllers may not provide satisfactory performance for highly nonlinear systems
• Techniques such as gain scheduling, adaptive control, or nonlinear control methods may be required to handle nonlinearities effectively

Time-delay systems

• Time delays introduce a phase lag between the control action and the system response
• PID controllers may have difficulty in controlling systems with significant time delays
• Time delays can lead to oscillations, instability, and poor control performance
• Specialized control techniques, such as Smith predictor or dead-time compensators, can be employed to address time delays

Noise sensitivity

• PID controllers, particularly the derivative term, are sensitive to noise and high-frequency disturbances in the measured process variable
• Noise can cause erratic control behavior and lead to excessive control action and actuator wear
• Filtering techniques, such as low-pass filters or signal smoothing, are often necessary to mitigate the effects of noise
• Careful tuning of the derivative gain and the use of derivative filters can help reduce noise sensitivity

PID controller applications

• PID controllers find widespread applications across various industries and domains
• Their simplicity, robustness, and effectiveness make them a popular choice for process control, motion control, and temperature regulation

Industrial process control

• PID controllers are extensively used in industrial process control applications
• Examples include chemical reactors, distillation columns, heat exchangers, and pressure vessels
• PID controllers regulate process variables such as temperature, pressure, flow rate, and level to maintain desired operating conditions
• Ensure product quality, safety, and efficiency in manufacturing processes

Motor speed control

• PID controllers are commonly used for motor speed control applications
• Regulate the speed of DC motors, AC motors, and servo motors
• Maintain constant speed under varying load conditions and disturbances
• Employed in robotics, machine tools, conveyor systems, and automotive applications

Temperature regulation

• PID controllers are widely used for temperature regulation in various applications
• Examples include HVAC systems, ovens, furnaces, and incubators
• Control heating and cooling elements to maintain a desired temperature setpoint
• Ensure precise temperature control for processes that require stable and uniform temperature conditions

PID controller variations

• Several variations of the standard PID controller are used to address specific control requirements or system characteristics
• These variations modify the structure or behavior of the PID algorithm to achieve improved performance or simplify the controller design

PI controller

• PI controller consists of only the proportional and integral terms
• Eliminates the derivative term, which can be sensitive to noise and high-frequency disturbances
• Suitable for systems with slow dynamics or where the derivative action is not necessary
• Provides good steady-state error elimination and disturbance rejection
• Commonly used in process control applications where the system response is not too fast

PD controller

• PD controller consists of only the proportional and derivative terms
• Eliminates the integral term, which can cause overshoot and oscillations in some systems
• Provides fast response and improved stability
• Suitable for systems with fast dynamics or where the steady-state error is not a concern
• Commonly used in motion control applications where quick response and damping are required

Two-degree-of-freedom PID

• Two-degree-of-freedom (2DOF) PID controller separates the setpoint tracking and disturbance rejection tasks
• Introduces additional parameters to independently tune the controller's response to setpoint changes and disturbances
• Allows for better control performance and flexibility in shaping the system response
• Particularly useful in systems where the setpoint and disturbance characteristics are different
• Provides improved setpoint tracking and disturbance rejection compared to the standard PID controller

PID controller implementation

• PID controllers can be implemented using various technologies and platforms, depending on the application requirements and available resources
• The choice of implementation depends on factors such as the system's complexity, response time, and integration with other control components

Analog PID controllers

• Analog PID controllers are implemented using electronic circuits and operational amplifiers
• Proportional, integral, and derivative actions are realized using resistors, capacitors, and other analog components
• Provide continuous control action and fast response times
• Suitable for systems with analog sensors and actuators
• Commonly used in standalone applications or as part of a larger analog control system

Digital PID controllers

• Digital PID controllers are implemented using microprocessors, microcontrollers, or digital signal processors (DSPs)
• PID algorithm is executed in software, using numerical methods for integration and differentiation
• Provide flexibility in terms of parameter tuning, data logging, and communication with other digital systems
• Suitable for systems with digital sensors and actuators or where integration with other digital components is required
• Commonly used in computer-based control systems, embedded systems, and industrial automation

PLC-based PID control

• Programmable Logic Controllers (PLCs) often include built-in PID control functionality
• PID algorithm is implemented as a function block or a dedicated PID instruction in the PLC programming language
• Provides seamless integration with other PLC-based control logic and I/O modules
• Suitable for industrial automation applications where PLCs are already used for process control and sequencing
• Offers advantages such as scalability, reliability, and ease of programming and maintenance