6.3 PID Control Theory and Implementation

PID control is a key component of closed-loop control systems. It uses proportional, integral, and derivative terms to minimize errors between desired and actual outputs. This versatile method is widely used in industrial processes for temperature, flow, and motion control.

Tuning PID controllers involves adjusting gains to optimize system response. Various methods like Ziegler-Nichols and Cohen-Coon help achieve desired performance. Stability analysis ensures robustness, while implementation requires discretization for digital systems and anti-windup techniques to prevent integral term issues.

PID Control Principles and Components

PID Control Fundamentals

Top images from around the web for PID Control Fundamentals

• 

  Control system - Wikipedia View original

  Is this image relevant?

• 

  File:Pid-feedback-nct-int-correct.png - Wikimedia Commons View original

  Is this image relevant?

• 

  PID controller - Wikipedia View original

  Is this image relevant?

• 

  Control system - Wikipedia View original

  Is this image relevant?

• 

  File:Pid-feedback-nct-int-correct.png - Wikimedia Commons View original

  Is this image relevant?

1 of 3

Top images from around the web for PID Control Fundamentals

• 

  Control system - Wikipedia View original

  Is this image relevant?

• 

  File:Pid-feedback-nct-int-correct.png - Wikimedia Commons View original

  Is this image relevant?

• 

  PID controller - Wikipedia View original

  Is this image relevant?

• 

  Control system - Wikipedia View original

  Is this image relevant?

• 

  File:Pid-feedback-nct-int-correct.png - Wikimedia Commons View original

  Is this image relevant?

1 of 3

• PID control is a feedback control mechanism that continuously calculates an error value as the difference between a desired setpoint (SP) and a measured process variable (PV) and applies a correction based on proportional, integral, and derivative terms
• PID controllers can be implemented in different forms, such as parallel, ideal, series, and discrete-time algorithms, each with its own advantages and limitations
• PID control is widely used in industrial processes, such as temperature control (heating and cooling systems), flow control (valves and pumps), and motion control (robotics and automation)
• The main objective of PID control is to minimize the error between the setpoint and the process variable, while ensuring stability, robustness, and optimal performance of the system

PID Control Terms and Their Effects

• The proportional term (P) produces an output value that is proportional to the current error value, providing a rapid correction to disturbances but may result in a steady-state error
  • Increasing the proportional gain (Kp) reduces the rise time and the steady-state error but may increase the overshoot and the oscillations
  • Example: In a temperature control system, a high proportional gain may cause the heater to quickly respond to temperature changes but may also cause the temperature to oscillate around the setpoint
• The integral term (I) produces an output value that is proportional to both the magnitude and duration of the error, eliminating the steady-state error but may cause the system to overshoot the setpoint and oscillate
  • Increasing the integral gain (Ki) eliminates the steady-state error and improves the disturbance rejection but may increase the overshoot and the settling time
  • Example: In a level control system, a high integral gain may ensure that the liquid level reaches the setpoint without any offset but may also cause the level to overshoot and oscillate before settling
• The derivative term (D) produces an output value that is proportional to the rate of change of the error, reducing overshoot and oscillation but is sensitive to noise and may cause instability
  • Increasing the derivative gain (Kd) reduces the overshoot and the settling time and improves the system stability but may increase the sensitivity to noise and the control effort
  • Example: In a speed control system, a high derivative gain may dampen the oscillations caused by sudden load changes but may also amplify the noise in the speed measurement signal

PID Controller Tuning for Performance

PID Tuning Methods and Techniques

• PID tuning involves adjusting the proportional, integral, and derivative gains (Kp, Ki, Kd) to achieve the desired system response, considering factors such as rise time, overshoot, settling time, and steady-state error
• The Ziegler-Nichols method is a heuristic PID tuning method that involves setting the integral and derivative gains to zero, increasing the proportional gain until the system oscillates with a constant amplitude, and then adjusting the gains based on the oscillation period and gain
  • The Ziegler-Nichols method provides a simple and quick way to tune the PID controller but may result in an aggressive and oscillatory response
  • Example: In a pressure control system, the Ziegler-Nichols method can be used to find the ultimate gain and period by gradually increasing the proportional gain until the system oscillates, and then calculating the PID gains based on the predefined formulas
• The Cohen-Coon method is another heuristic PID tuning method that is based on the open-loop step response of the system and provides a set of equations to calculate the PID gains based on the process characteristics
  • The Cohen-Coon method is more suitable for processes with a large dead time and a small time constant but may require a process model or a step test to determine the process parameters
  • Example: In a flow control system, the Cohen-Coon method can be used to estimate the process gain, dead time, and time constant from the open-loop step response, and then calculate the PID gains based on the provided equations
• Model-based PID tuning methods, such as the Internal Model Control (IMC) and the Lambda tuning method, use a mathematical model of the process to calculate the optimal PID gains based on the desired closed-loop response
  • Model-based PID tuning methods provide a systematic and optimal way to tune the PID controller but may require a precise and accurate process model and may be sensitive to model uncertainties and disturbances
  • Example: In a chemical reactor control system, the IMC method can be used to derive the ideal PID controller based on the inverse of the process model and a low-pass filter, and then approximate the ideal controller with a realizable PID controller
• Adaptive PID tuning methods, such as the Gain Scheduling and the Self-Tuning Regulator (STR), automatically adjust the PID gains based on the changing process conditions or the estimated process model
  • Adaptive PID tuning methods provide a flexible and robust way to tune the PID controller but may require additional sensors, computations, and algorithms to estimate the process parameters and adapt the controller gains
  • Example: In a wind turbine control system, the Gain Scheduling method can be used to adjust the PID gains based on the wind speed and the turbine operating point, using a predefined set of gain values for different operating regions

PID Stability Analysis and Robustness

• Stability analysis techniques, such as the Routh-Hurwitz criterion and the Nyquist stability criterion, can be used to determine the stability of the closed-loop system and ensure that the PID gains are within the stable region
  • The Routh-Hurwitz criterion provides a necessary and sufficient condition for the stability of a linear time-invariant system based on the coefficients of its characteristic equation
  • Example: In a DC motor control system, the Routh-Hurwitz criterion can be used to find the range of the PID gains that guarantee the stability of the closed-loop system, by analyzing the coefficients of the characteristic equation and ensuring that all the elements in the first column of the Routh array are positive
• The Nyquist stability criterion provides a graphical method to determine the stability of a closed-loop system based on the open-loop frequency response and the number of encirclements of the critical point (-1, 0) by the Nyquist plot
  • The Nyquist stability criterion can also be used to determine the gain and phase margins of the system, which indicate the amount of gain and phase shift that the system can tolerate before becoming unstable
  • Example: In a spacecraft attitude control system, the Nyquist stability criterion can be used to analyze the stability and robustness of the PID controller, by plotting the open-loop frequency response and ensuring that the Nyquist plot does not encircle the critical point and has sufficient gain and phase margins
• Robustness analysis techniques, such as the sensitivity function and the complementary sensitivity function, can be used to evaluate the ability of the PID controller to reject disturbances, follow setpoint changes, and tolerate model uncertainties
  • The sensitivity function represents the ratio of the output disturbance to the input disturbance and should be small at low frequencies to ensure good disturbance rejection and setpoint tracking
  • The complementary sensitivity function represents the ratio of the output to the setpoint and should be small at high frequencies to ensure good noise attenuation and robustness to model uncertainties
  • Example: In a hydraulic servo system, the sensitivity and complementary sensitivity functions can be used to design a robust PID controller that minimizes the effect of load disturbances, sensor noise, and parameter variations on the system performance, by shaping the frequency response of the closed-loop system using weighting functions and optimization techniques

PID Control Algorithm Implementation

PID Algorithm Discretization and Programming

• PID control algorithms can be implemented in software using programming languages such as C, C++, Python, or MATLAB, either as a standalone application or as part of a larger control system

• The PID algorithm can be discretized using numerical integration methods, such as the forward Euler, backward Euler, or trapezoidal rule, to approximate the continuous-time integral and derivative terms

  • The forward Euler method is the simplest and most computationally efficient but may introduce numerical errors and instability, especially for large sampling times
  • The backward Euler method is more stable and accurate but may require solving a system of equations and may introduce a delay in the control action
  • The trapezoidal rule provides a good balance between accuracy and stability but may require more computations and memory than the Euler methods
  • Example: In a temperature control system implemented in C++, the PID algorithm can be discretized using the trapezoidal rule as follows:

    double error = setpoint - measurement;
    integral += (error + prevError) * dt / 2;
    double derivative = (error - prevError) / dt;
    double output = Kp * error + Ki * integral + Kd * derivative;
    prevError = error;
    

• Anti-windup techniques, such as the back-calculation and the clamping method, can be used to prevent the integral term from accumulating error when the actuator is saturated, causing the system to overshoot or oscillate

  • The back-calculation method reduces the integral term by a factor proportional to the difference between the saturated and unsaturated control output, effectively reducing the integral action when the actuator is saturated
  • The clamping method limits the integral term to a predefined range, preventing it from accumulating beyond the actuator limits and causing windup
  • Example: In a pressure control system implemented in Python, the PID algorithm can be modified with the back-calculation anti-windup method as follows:

    error = setpoint - measurement
    integral += error * dt
    if output > max_output:
        integral -= (output - max_output) / Ki
    elif output < min_output:
        integral -= (output - min_output) / Ki
    derivative = (error - prev_error) / dt
    output = Kp * error + Ki * integral + Kd * derivative
    prev_error = error
    

• Bumpless transfer techniques, such as the setpoint weighting and the output tracking method, can be used to smoothly transition between manual and automatic control modes without causing a sudden change in the control signal

  • The setpoint weighting method filters the setpoint change using a first-order lag filter, reducing the proportional and derivative kicks during mode transitions

  • The output tracking method adjusts the integral term to match the manual output before switching to automatic mode, ensuring a smooth and continuous control action

  • Example: In a flow control system implemented in MATLAB, the PID algorithm can be modified with the setpoint weighting bumpless transfer method as follows:

    error = beta * setpoint - measurement;
    integral = integral + Ki * error * dt;
    derivative = Kd * (error - prevError) / dt;
    output = Kp * error + integral + derivative;
    prevError = error;
    

    where

    beta

    is the setpoint weighting factor between 0 and 1, with 0 corresponding to a pure PI controller and 1 corresponding to a pure PID controller.

PID Hardware Implementation and Communication

• PID control can be implemented in hardware using analog or digital circuits, such as operational amplifiers, microcontrollers, or programmable logic controllers (PLCs), depending on the application requirements
  • Analog PID controllers use operational amplifiers and passive components, such as resistors and capacitors, to realize the PID algorithm in continuous time, providing fast and smooth control action but limited flexibility and functionality
  • Digital PID controllers use microcontrollers or digital signal processors (DSPs) to execute the PID algorithm in discrete time, providing more flexibility, programmability, and advanced features but requiring analog-to-digital and digital-to-analog conversions and introducing sampling and quantization effects
  • PLCs are specialized industrial computers that execute the PID algorithm and other control logic in a cyclic manner, using a programming language such as Ladder Diagram or Structured Text, and providing a rugged and modular hardware platform for process control applications
  • Example: In a chemical dosing system, a PLC-based PID controller can be used to regulate the flow rate of a dosing pump based on the pH measurement of a tank, using a 4-20 mA current loop for signal transmission and a PWM output for pump control
• Fieldbus communication protocols, such as HART, PROFIBUS, or Foundation Fieldbus, can be used to transmit the PID control signals and process measurements between the controller and the field devices
  • Fieldbus protocols provide a digital, bidirectional, and multi-drop communication network that allows the controller to access the device parameters, diagnostics, and status information, and the devices to execute complex control algorithms and transmit multiple process variables
  • Fieldbus protocols use a variety of physical layers, such as twisted-pair, coaxial cable, or fiber optic, and a variety of data link and application layers, such as master-slave, token-passing, or client-server, depending on the specific protocol and application
  • Example: In a gas pipeline control system, a Foundation Fieldbus-based PID controller can be used to regulate the pressure and flow rate of the gas using a set of pressure and flow transmitters and control valves, communicating over a single twisted-pair cable and executing the PID algorithm in the field devices

PID Parameter Effects on System Response

Effects of PID Gains on System Performance

• The effects of the proportional gain (Kp) on the system response include reducing the rise time and the steady-state error, but increasing the overshoot and the settling time, and may cause the system to become unstable if set too high
  • A high proportional gain provides a fast and aggressive control action but may cause the system to oscillate or even become unstable, especially in the presence of delays, nonlinearities, or uncertainties
  • A low proportional gain provides a slow and gentle control action but may result in a large steady-state error and a poor disturbance rejection, especially for processes with a large dead time or a small gain
  • Example: In a cruise control system for a car, increasing the proportional gain of the PID controller may improve the speed tracking performance and the responsiveness to changes in the road grade or the wind speed but may also cause the car to accelerate and decelerate abruptly and oscillate around the setpoint
• The effects of the integral gain (Ki) on the system response include eliminating the steady-state error and improving the disturbance rejection, but increasing the overshoot and the settling time, and may cause the system to oscillate if set too high
  • A high integral gain provides a fast and complete elimination of the steady-state error and a good disturbance rejection but may cause the system to overshoot and oscillate, especially for processes with a large time constant or a small gain
  • A low integral gain provides a slow and partial elimination of the steady-state error and a poor disturbance rejection but may improve the stability and the damping of the system, especially for processes with a small time constant or a large gain
  • Example: In a water level control system for a tank, increasing the integral gain of the PID controller may ensure that the water level reaches the setpoint without any offset and reject the disturbances caused by the inflow or the outflow changes but may also cause the water level to overshoot and oscillate before settling, especially if the tank has a large cross-sectional area or a small outlet valve
• The effects of the derivative gain (Kd) on the system response include reducing the overshoot and the settling time, and improving the system stability, but increasing the sensitivity to noise and may cause the system to become unstable if set too high
  • A high derivative gain provides a fast and anticipatory control action that predicts and counteracts the changes in the error signal but may amplify the noise and the high-frequency disturbances, causing the system to chatter or oscillate
  • A low derivative gain provides a slow and reactive control action that responds only to the current error signal but may reduce the noise sensitivity and the control effort, improving the system robustness and the actuator life
  • Example: In a robotic arm position control system, increasing the derivative gain of the PID controller may reduce the overshoot and the settling time of the arm position and improve the tracking of the reference trajectory but may also cause the arm to vibrate or jitter due to the noise in the position sensor or the torque ripple in the motor

Effects of Sampling Time and Process Dynamics on PID Control

• The effects of the sampling time on the system response include improving the control accuracy and the disturbance rejection, but increasing the computational burden and the sensitivity to noise, and may cause the system to become unstable if set too low
  • A high sampling rate provides a more frequent and accurate measurement of the process variable and a faster update of the control output but may increase the processor load, the communication bandwidth, and the data storage requirements, especially for processes with a large number of inputs and outputs
  • A low sampling rate provides a less frequent and accurate measurement of the process variable and a slower update of the control output but may reduce the processor load, the communication bandwidth, and the data storage requirements, especially for processes with a small number of inputs and outputs
  • Example: In a temperature control system for a furnace, increasing the sampling rate of the PID controller may improve the temperature regulation and the rejection of the load disturbances but may also increase the noise in the temperature measurement and the wear of the heating elements, especially if the furnace has a large thermal mass or a small heating power
• The effects of the actuator and sensor dynamics on the system response include introducing delays, nonlinearities, and uncertainties that can degrade the control performance and the stability, and may require advanced compensation techniques, such as feedforward, cascade, or model predictive control
  • Actuator dynamics, such as the hysteresis, the backlash, the saturation, and the rate limit