4.5 PID control

PID control is a crucial technique in autonomous robotics, helping maintain desired setpoints by adjusting control inputs based on measured errors. It uses proportional, integral, and derivative terms to generate control signals, balancing rapid response, steady-state error elimination, and system stability.

PID controllers are widely applied in motor speed control, temperature regulation, and vehicle steering. While effective, they have limitations like integral windup and derivative kick. Variations like PI control and feedforward control can address these issues, improving overall system performance and robustness.

PID control fundamentals

• PID control is a widely used feedback control technique in autonomous robots and other systems to maintain a desired setpoint by adjusting the control input based on the measured error
• PID stands for Proportional-Integral-Derivative, which are the three main components of the controller that generate the control signal based on the error between the desired setpoint and the measured process variable

Proportional, integral, derivative terms

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• The proportional term ($K_p$) produces a control output proportional to the current error, providing a rapid response to reduce the error but may lead to steady-state error
• The integral term ($K_i$) accumulates the error over time and generates a control output to eliminate the steady-state error, but can cause overshoot and oscillations if not tuned properly
• The derivative term ($K_d$) responds to the rate of change of the error, helping to dampen the system response and improve stability, but is sensitive to noise and can cause large control outputs for sudden changes in error

Feedback control loop

• In a PID feedback control loop, the controller continuously compares the measured process variable (PV) with the desired setpoint (SP) to calculate the error signal
• The PID controller then generates a control output based on the proportional, integral, and derivative terms of the error signal
• This control output is applied to the system (plant) to adjust its behavior and drive the process variable towards the setpoint

Error signal calculation

• The error signal, $e(t)$, is calculated as the difference between the desired setpoint, $r(t)$, and the measured process variable, $y(t)$: $e(t) = r(t) - y(t)$
• The goal of the PID controller is to minimize this error signal over time by generating an appropriate control output

Setpoint vs process variable

• The setpoint (SP) is the desired value or target that the system should maintain, such as a desired motor speed (1500 RPM) or temperature (50°C)
• The process variable (PV) is the actual measured value of the system output, obtained through sensors or feedback devices (encoder, thermocouple)
• The PID controller aims to make the process variable track the setpoint closely, minimizing the difference between them (error signal)

PID controller design

• Designing a PID controller involves selecting the appropriate gains ($K_p$, $K_i$, $K_d$) to achieve the desired system performance, such as fast response, minimal overshoot, and good disturbance rejection
• The PID controller can be represented using a transfer function in the Laplace domain, which relates the control output to the error signal

Transfer function representation

• The transfer function of a PID controller, $G_c(s)$, is given by: $G_c(s) = K_p + \frac{K_i}{s} + K_d s$ where $s$ is the Laplace variable
• This transfer function can be used to analyze the system's stability, performance, and robustness using tools like root locus, Bode plots, and Nyquist diagrams

Tuning PID gains

• Tuning the PID gains ($K_p$, $K_i$, $K_d$) is crucial for achieving optimal controller performance and stability
• The gains can be adjusted iteratively based on the system's response to step inputs or disturbances until the desired performance is achieved

Manual tuning methods

• Manual tuning involves adjusting the gains based on the observed system response and following general guidelines:
  1. Start with $K_i = K_d = 0$ and increase $K_p$ until the system oscillates
  2. Increase $K_i$ to reduce steady-state error and oscillations
  3. Increase $K_d$ to improve stability and reduce overshoot
• Fine-tune the gains iteratively to achieve the desired performance (quarter-decay ratio, 10-20% overshoot)

Ziegler-Nichols tuning

• The Ziegler-Nichols method is a systematic approach to tune PID gains based on the system's critical gain ($K_{cr}$) and critical period ($T_{cr}$)
• The critical gain is the value of $K_p$ that causes the system to oscillate with a constant amplitude, and the critical period is the oscillation period at this gain
• The PID gains are then calculated using predefined formulas based on $K_{cr}$ and $T_{cr}$:
  • $K_p = 0.6 K_{cr}$
  • $K_i = 2 K_p / T_{cr}$
  • $K_d = K_p T_{cr} / 8$

Simulink modeling of PID

• Simulink, a graphical programming environment in MATLAB, can be used to model and simulate PID control systems
• PID controllers can be represented using built-in Simulink blocks (PID Controller) or custom subsystems with individual P, I, and D blocks
• Simulink allows easy integration of plant models, sensors, and actuators to test and optimize the PID controller performance before implementation

Discrete vs continuous time

• PID controllers can be implemented in either continuous time (analog) or discrete time (digital) domains
• Continuous-time PID controllers operate on analog signals and are described using differential equations or transfer functions in the Laplace domain
• Discrete-time PID controllers operate on sampled signals and are described using difference equations or transfer functions in the Z-domain
• When implementing PID control on digital systems (microcontrollers, computers), the continuous-time controller must be discretized using techniques like Euler, Tustin, or matched pole-zero methods

PID control applications

• PID control is widely used in various applications in autonomous robots and other systems where precise control and setpoint tracking are required

Motor speed control

• PID controllers can be used to regulate the speed of DC motors or brushless DC motors (BLDC) in robotic applications
• The setpoint is the desired motor speed (RPM), and the process variable is the actual speed measured using encoders or hall sensors
• The PID controller adjusts the motor drive signal (PWM) based on the speed error to maintain the desired speed under varying loads or disturbances

Temperature regulation

• PID control is commonly used in temperature regulation systems, such as heat exchangers, ovens, or thermal chambers
• The setpoint is the desired temperature (°C or °F), and the process variable is the actual temperature measured using thermocouples or RTDs
• The PID controller adjusts the heating or cooling power (current, voltage) based on the temperature error to maintain the desired temperature profile

Autonomous vehicle steering

• PID controllers can be used for lateral control of autonomous vehicles, such as cars or mobile robots, to follow a desired path or maintain lane position
• The setpoint is the desired steering angle or lateral position, and the process variable is the actual steering angle (measured by encoders) or lateral position (measured by cameras or GPS)
• The PID controller generates the steering command based on the error signal to minimize the deviation from the desired path

Drone altitude stabilization

• PID control is used in drone altitude stabilization systems to maintain a constant height or follow a desired trajectory
• The setpoint is the desired altitude (meters), and the process variable is the actual altitude measured using barometers, ultrasonic sensors, or GPS
• The PID controller adjusts the thrust command to the propellers based on the altitude error to counteract disturbances (wind) and maintain stable flight

PID limitations and variations

• While PID control is effective in many applications, it has some limitations and challenges that can be addressed using variations or extensions of the basic PID structure

Integral windup

• Integral windup occurs when the integral term accumulates a large error during periods of actuator saturation or when the system cannot respond quickly to changes in setpoint
• This leads to overshoots, oscillations, and slow recovery when the error changes sign
• Anti-windup techniques, such as conditional integration or tracking, can be used to mitigate integral windup by limiting the integral term or resetting it based on the actuator saturation status

Derivative kick

• Derivative kick is a sudden large control output that occurs when there is an abrupt change in the setpoint, causing a spike in the error signal
• This can lead to actuator saturation and system instability
• Setpoint weighting or filtering the derivative term can help reduce the impact of derivative kick by smoothing the setpoint changes or using the process variable instead of the error for the derivative action

PI vs PID control

• In some applications, the derivative term may not be necessary or can amplify noise, leading to poor performance
• PI control, which uses only the proportional and integral terms, can be sufficient for systems with slow dynamics or low noise levels
• PI control is easier to tune and implement than PID control and can provide good setpoint tracking and disturbance rejection in many cases

Feedforward control

• Feedforward control is a technique that uses knowledge of the system model and expected disturbances to generate a control output that proactively compensates for these disturbances
• Feedforward control can be combined with PID control to improve the system's response to known disturbances (load changes, friction) and reduce the burden on the feedback controller
• Feedforward control requires an accurate system model and disturbance estimation, which can be challenging in practice

Cascade PID control

• Cascade PID control is a multi-loop control strategy that uses two or more PID controllers in a hierarchical structure to improve the system's performance and robustness
• The outer loop controller generates the setpoint for the inner loop controller, which directly controls the actuator
• Cascade control can help to reject disturbances and improve the system's response time by leveraging the faster dynamics of the inner loop (motor current control) compared to the outer loop (position control)

PID robustness and performance

• PID controllers must be designed and tuned to ensure robustness and performance in the presence of uncertainties, disturbances, and nonlinearities

Disturbance rejection

• Disturbance rejection is the ability of the PID controller to minimize the impact of external disturbances on the system's output
• Good disturbance rejection is achieved by proper tuning of the PID gains, especially the integral and derivative terms, which help to quickly eliminate the steady-state error and dampen the system's response
• Disturbance rejection can be quantified using metrics like the sensitivity function or the integral of absolute error (IAE)

Steady-state error

• Steady-state error is the difference between the setpoint and the process variable when the system reaches a stable equilibrium
• The integral term in the PID controller is responsible for eliminating the steady-state error by accumulating the error over time and generating a corrective control output
• The steady-state error can be reduced by increasing the integral gain ($K_i$), but this may lead to overshoot and oscillations if not balanced with the other gains

Rise time and settling time

• Rise time is the time required for the system's output to reach a specified percentage (usually 90%) of its final value when responding to a step input
• Settling time is the time required for the system's output to settle within a specified tolerance band (usually ±2% or ±5%) around its final value
• PID controllers can be tuned to achieve a desired rise time and settling time by adjusting the proportional and derivative gains ($K_p$, $K_d$), which influence the system's responsiveness and damping

Overshoot and oscillations

• Overshoot is the percentage by which the system's output exceeds its final value during the transient response to a step input
• Oscillations are the periodic fluctuations of the system's output around its final value, which can be caused by insufficient damping or improper gain tuning
• PID controllers can be tuned to minimize overshoot and oscillations by adjusting the derivative gain ($K_d$) and the integral gain ($K_i$), which provide damping and reduce the steady-state error, respectively

Stability analysis

• Stability analysis is the study of the PID control system's ability to maintain bounded outputs for bounded inputs and initial conditions
• Stability can be assessed using tools like root locus, Bode plots, and Nyquist diagrams, which provide insights into the system's poles, zeros, gain and phase margins
• A stable PID control system should have all its poles in the left-half of the complex plane (continuous time) or inside the unit circle (discrete time) and sufficient gain and phase margins to tolerate uncertainties and delays

PID implementation considerations

• When implementing PID controllers in real-world systems, several practical considerations must be taken into account to ensure reliable and efficient operation

Analog vs digital controllers

• PID controllers can be implemented using analog electronic circuits (operational amplifiers, resistors, capacitors) or digital processors (microcontrollers, FPGAs, PLCs)
• Analog PID controllers offer fast response times and continuous control signals but are more susceptible to noise, drift, and component variations
• Digital PID controllers provide flexibility, programmability, and integration with other control algorithms but introduce quantization errors, sampling delays, and computational limitations

Sensor noise filtering

• Sensor noise can introduce high-frequency fluctuations in the measured process variable, which can degrade the PID controller's performance and stability, especially the derivative term
• Low-pass filters (analog or digital) can be used to attenuate the high-frequency noise components and smooth the sensor signal before feeding it to the PID controller
• The filter's cutoff frequency should be selected based on the system's bandwidth and the noise characteristics to avoid excessive phase lag or signal attenuation

Actuator saturation

• Actuator saturation occurs when the control output demanded by the PID controller exceeds the physical limits of the actuator (maximum voltage, current, or force)
• Saturation can lead to integral windup, loss of control authority, and system instability if not properly handled
• Anti-windup techniques, such as conditional integration or tracking, should be implemented to prevent the integral term from accumulating excessive error during saturation and ensure smooth recovery when the actuator becomes unsaturated

Sampling rate selection

• In digital PID controllers, the sampling rate (or control loop frequency) determines how often the controller updates its output based on the measured error signal
• The sampling rate should be at least 10-20 times faster than the system's dominant time constant or closed-loop bandwidth to avoid aliasing and ensure adequate control performance
• Oversampling can improve the controller's resolution and noise rejection but increases the computational burden and may introduce unnecessary delays

Code optimization techniques

• When implementing PID controllers on resource-constrained embedded systems, code optimization techniques can help to reduce the computational overhead and memory usage
• Fixed-point arithmetic can be used instead of floating-point to speed up calculations and reduce memory footprint, but care must be taken to avoid overflow and precision loss
• Lookup tables can be used to store pre-computed values of trigonometric, exponential, or logarithmic functions to avoid expensive runtime calculations
• Inline functions and compiler optimizations (loop unrolling, constant folding) can be leveraged to improve the execution speed and reduce the code size