PID Control

PID (Proportional-Integral-Derivative) control is a widely used control strategy in industrial automation for maintaining desired levels of process variables such as temperature, pressure, flow, and speed.

Understanding PID Control

PID control is a feedback control loop mechanism that calculates the error value as the difference between a measured process variable and a desired setpoint. The controller attempts to minimize this error by adjusting the process control inputs. The PID controller consists of three terms: Proportional (P), Integral (I), and Derivative (D).

• Proportional Term (P)
• Integral Term (I)
• Derivative Term (D)
• PID Control Algorithm
• Ziegler-Nichols Method for PID Tuning
• Trial and Error Method for PID Tuning

Proportional Term (P)

The proportional term produces an output value that is proportional to the current error value. It provides a control action that is directly proportional to the error.

Formula: P = K_p × e(t)

• P: Proportional term output
• K_p: Proportional gain
• e(t): Error at time t

Proportional Gain (K_p):

Error (e(t)):

Calculate Proportional Term

Integral Term (I)

The integral term produces an output value that is proportional to the cumulative sum of past error values. It helps eliminate residual steady-state error by integrating the error over time.

Formula: I = K_i × ∫e(t) dt

• I: Integral term output
• K_i: Integral gain
• ∫e(t) dt: Integral of the error over time

Integral Gain (K_i):

Integral of Error (∫e(t) dt):

Calculate Integral Term

Derivative Term (D)

The derivative term produces an output value that is proportional to the rate of change of the error. It predicts future error based on its rate of change and helps reduce overshoot and oscillations.

Formula: D = K_d × (de(t)/dt)

• D: Derivative term output
• K_d: Derivative gain
• de(t)/dt: Rate of change of error

Derivative Gain (K_d):

Rate of Change of Error (de(t)/dt):

Calculate Derivative Term

PID Control Algorithm

The PID control algorithm combines the three terms to calculate the control output. The overall PID control output is given by the sum of the proportional, integral, and derivative terms.

Formula: u(t) = K_p × e(t) + K_i × ∫e(t) dt + K_d × (de(t)/dt)

• u(t): Control output
• K_p: Proportional gain
• K_i: Integral gain
• K_d: Derivative gain
• e(t): Error at time t
• ∫e(t) dt: Integral of the error over time
• de(t)/dt: Rate of change of error

Proportional Gain (K_p):

Integral Gain (K_i):

Derivative Gain (K_d):

Error (e(t)):

Integral of Error (∫e(t) dt):

Derivative of Error (de(t)/dt):

Calculate Control Output

Example Calculation

Consider a PID controller with the following parameters:

• Proportional Gain (K_p): 2
• Integral Gain (K_i): 1
• Derivative Gain (K_d): 0.5
• Current Error (e(t)): 5
• Integral of Error (∫e(t) dt): 10
• Rate of Change of Error (de(t)/dt): 2

Solution: Use the PID control formula to calculate the control output:

• Proportional Term: P = K_p × e(t) = 2 × 5 = 10
• Integral Term: I = K_i × ∫e(t) dt = 1 × 10 = 10
• Derivative Term: D = K_d × (de(t)/dt) = 0.5 × 2 = 1
• Total Control Output: u(t) = P + I + D = 10 + 10 + 1 = 21

Ziegler-Nichols Method for PID Tuning

The Ziegler-Nichols method is a widely used heuristic technique for tuning the parameters of a PID (Proportional-Integral-Derivative) controller. It provides a set of guidelines to determine the optimal PID controller parameters (proportional, integral, and derivative gains) based on the system's response to a step input.

Steps in the Ziegler-Nichols Method:

1. Open-Loop Test

   Perform an open-loop test by applying a step input to the system and measuring the system's response. This is done without any feedback control to observe the system's natural behavior.

2. Increase the Proportional Gain (Kₚ)

   Gradually increase the proportional gain (Kₚ) and observe the system's response. The goal is to find the critical gain (Kₐ) where the system oscillates with a constant amplitude.

   Critical Gain (Kₐ): The value of Kₚ at which the system oscillates with constant amplitude.

   Oscillation Period (Pₛ): The time taken to complete one full oscillation at Kₐ.

3. Calculate the PID Parameters

   Once the critical gain (Kₐ) and oscillation period (Pₛ) are determined, use the following formulas to calculate the PID controller gains:

   • Proportional Gain (Kₚ): Kₚ = 0.6 × Kₐ
   • Integral Gain (Kᵢ): Kᵢ = 2 × Kₚ / Pₛ
   • Derivative Gain (Kᵈ): Kᵈ = Kₚ × Pₛ / 8

   These values are the starting point for the PID controller, and fine-tuning may be needed for optimal performance.

4. Apply the PID Controller

   Use the calculated PID gains in the system's controller. Observe the system's performance and adjust the gains if necessary to achieve the desired system behavior.

Example Calculation:

Suppose you are controlling a heating system and apply a step input to the system. The system reaches oscillation at a critical gain Kₐ = 10, and the oscillation period Pₛ = 2 seconds. Using the Ziegler-Nichols formulas, you can calculate the PID parameters as follows:

• Kₚ = 0.6 × 10 = 6
• Kᵢ = 2 × 6 / 2 = 6
• Kᵈ = 6 × 2 / 8 = 1.5

Advantages of Ziegler-Nichols Method:

• Quick and Simple: The method is simple and provides a quick way to estimate PID controller gains, especially in systems where dynamics are hard to model.
• Good Starting Point: It provides a reasonable starting point for PID tuning, which can then be fine-tuned as necessary.

Limitations:

• May Not Be Optimal: The method may not always provide the best performance, especially for systems with nonlinearities or unique behaviors.
• Oscillatory Response: The method typically results in a controller that has oscillations, so additional fine-tuning may be required to reduce overshoot or settling time.

Trial and Error Method for PID Tuning

The Trial and Error method is one of the simplest and most intuitive ways to tune PID controllers. This method involves manually adjusting the PID gains while observing the system's response. The goal is to find the best combination of PID parameters that achieve the desired system performance, such as minimizing overshoot or settling time.

Steps in the Trial and Error Method:

1. Start with Proportional Gain (K_p)

   Begin by setting the proportional gain (K_p) to a small value. Gradually increase the gain until the system starts responding to the error. A higher proportional gain leads to a faster response, but too high a value can lead to overshoot or oscillation.

2. Adjust Integral Gain (K_i)

   Next, adjust the integral gain (K_i). Start with a low value and increase it if the system has steady-state error (e.g., if it does not reach the setpoint after a long period). Be cautious, as too much integral gain can cause oscillation or slow response.

3. Fine-Tune Derivative Gain (K_d)

   After adjusting the proportional and integral gains, start modifying the derivative gain (K_d). The derivative gain helps reduce overshoot and damping oscillations. Start with a low value and increase it gradually to improve stability without introducing too much lag.

4. Iterative Adjustments

   Repeat the process of adjusting the PID gains, focusing on minimizing overshoot, settling time, and steady-state error. This may require several iterations as the effects of each gain can interact with one another.

Advantages of Trial and Error Method:

• Simple and Intuitive: No need for complex calculations or system modeling.
• Quick to Implement: Can be done in real-time without needing advanced tools.
• Good for Unknown Systems: Useful when the system's dynamics are not well understood.

Limitations of Trial and Error Method:

• Time-Consuming: Requires multiple iterations to fine-tune the parameters.
• Less Precise: May not always yield the optimal parameters, especially for complex systems.
• Risk of Over-Adjusting: It’s easy to introduce oscillations or instability by adjusting the parameters too aggressively.

Example:

Imagine you are tuning a heating system's PID controller:

• Start by setting the K_p to a small value, say 1. Observe the system’s response.
• If the system responds too slowly, increaseK_p to 3.
• Adjust K_i to reduce steady-state error. If the system is still not reaching the setpoint, increase K_i incrementally.
• Finally, adjust K_d to minimize overshoot and dampen oscillations. Start with a low value and increase it to 0.5.

Through repeated adjustments, you can find the PID settings that give the best performance for your system.