PID Calculation Formula in PLC | Control Equation, Tuning & Programming Guide

What is PID Control?

PID control is a closed-loop feedback mechanism used to maintain a process variable (PV) at a desired setpoint (SP). It continuously calculates an error value as the difference between the SP and PV and applies a correction based on proportional, integral, and derivative terms. The plc pid control formula combines these terms to generate a control output that drives the process towards the desired setpoint.

The PID Control Equation

The pid control equation plc is the mathematical foundation of PID control. It combines the proportional, integral, and derivative terms to calculate the control output (CO). The general form of the PID equation is:

CO(t) = K_p * e(t) + K_i * ∫e(t)dt + K_d * de(t)/dt

Where:

• CO(t) is the control output at time t
• K_p is the proportional gain
• K_i is the integral gain
• K_d is the derivative gain
• e(t) is the error signal (SP - PV) at time t
• ∫e(t)dt is the integral of the error signal over time
• de(t)/dt is the rate of change of the error signal over time

Understanding the Terms

1. Proportional (P) Term: The proportional term provides an immediate response to the error. A higher K_p results in a larger correction for a given error. However, too high a K_p can lead to oscillations.
2. Integral (I) Term: The integral term eliminates steady-state errors by accumulating the error over time. A higher K_i eliminates the error faster, but too high a K_i can cause instability.
3. Derivative (D) Term: The derivative term anticipates future errors by responding to the rate of change of the error. A higher K_d dampens oscillations, but too high a K_d can make the system sensitive to noise.

Discrete-Time PID Formula

In a PLC, the PID equation is typically implemented in discrete time. The discrete-time PID formula looks like this:

CO[n] = K_p * e[n] + K_i * T * Σe[i] + K_d * (e[n] - e[n-1]) / T

Where:

• CO[n] is the control output at the current sample time n
• e[n] is the error signal at the current sample time n
• e[n-1] is the error signal at the previous sample time n-1
• T is the sampling time
• Σe[i] is the sum of the error signals from i=0 to n

Here's an example of how this might look in PLC code (structured text):

// Inputs
REAL Setpoint;
REAL ProcessVariable;
REAL Kp;
REAL Ki;
REAL Kd;
REAL SampleTime;

// Outputs
REAL ControlOutput;

// Internal Variables
REAL Error;
REAL IntegralTerm;
REAL DerivativeTerm;
REAL PreviousError;

// Calculate Error
Error := Setpoint - ProcessVariable;

// Calculate Integral Term
IntegralTerm := IntegralTerm + (Error * SampleTime * Ki);

// Calculate Derivative Term
DerivativeTerm := (Error - PreviousError) / SampleTime * Kd;

// Calculate Control Output
ControlOutput := (Kp * Error) + IntegralTerm + DerivativeTerm;

// Update Previous Error
PreviousError := Error;

PID Tuning Methods in PLC

Tuning a PID controller involves finding the optimal values for K_p, K_i, and K_d that provide stable and responsive control. Several pid tuning methods in plc are available, each with its own advantages and disadvantages.

Common Tuning Methods

1. Trial and Error: This method involves manually adjusting the gains while observing the system's response. It's simple but can be time-consuming and may not yield optimal results.
2. Ziegler-Nichols Method: This method involves increasing K_p until the system oscillates continuously, then using the oscillation period and gain to calculate the PID gains.
3. Cohen-Coon Method: Similar to Ziegler-Nichols, but uses a step response to determine the process parameters.
4. Automatic Tuning (Auto-tuning): Many PLCs offer built-in auto-tuning functions that automatically calculate the PID gains based on the system's response to a test signal.

Here's a table comparing the different tuning methods:

The pid tuning formula in plc varies depending on the method used, but generally involves calculating the gain values based on observed process behavior.

Visualizing PID Tuning

The following chart illustrates the effect of different PID parameters on the system response. It demonstrates how adjusting the gains impacts overshoot, settling time, and steady-state error.

PLC PID Programming

Most PLC manufacturers provide built-in PID function blocks or instructions that simplify the implementation of PID control. These function blocks typically handle the calculations and logic required for PID control, allowing programmers to focus on configuring the parameters and integrating the controller into the overall system.

The pid function block plc usually has inputs for setpoint, process variable, and PID gains, and an output for the control output. Some function blocks also include advanced features such as anti-windup, bumpless transfer, and gain scheduling.

Example using Siemens TIA Portal

In Siemens TIA Portal, the PID Compact instruction is commonly used. Here's a simplified example of how to configure and use it:

// Create a PID Compact instance
"PID_Compact_DB"(
    Setpoint := Setpoint,
    ProcessValue := ProcessValue,
    Kp := Kp,
    Ti := Ti,  // Integral Time (in seconds)
    Td := Td,  // Derivative Time (in seconds)
    Man_On := ManualMode,
    ManValue := ManualValue,
    En := EnablePID,
    Out => ControlOutput
);

This code snippet shows how to connect the setpoint, process variable, and PID gains to the PID Compact function block. The `Ti` and `Td` parameters represent the integral and derivative times, respectively, which are related to the integral and derivative gains by the following equations:

• K_i = K_p / T_i
• K_d = K_p * T_d

PID Control Examples in PLC

PID control is used extensively in various industrial applications. Here are a few examples:

1. Temperature Control: Maintaining the temperature of a reactor vessel by adjusting the flow of coolant.
2. Flow Control: Regulating the flow rate of a liquid or gas in a pipeline by adjusting a control valve.
3. Pressure Control: Maintaining the pressure in a tank or vessel by adjusting a valve or pump.
4. Level Control: Controlling the liquid level in a tank by adjusting the inflow or outflow rate.

Example: Water Tank Level Control

Consider a water tank where the goal is to maintain a specific water level. A PID controller can be used to adjust the inflow rate to match the outflow rate and keep the level at the desired setpoint.

Conclusion

Understanding the PID calculation formula in PLC is essential for anyone working with automated control systems. By mastering the PID algorithm, its components, and tuning methods, you can design and implement robust and efficient control loops that optimize industrial processes. Explore your PLC's built-in PID function blocks and experiment with different tuning methods to gain practical experience. Ready to take your PLC skills to the next level? Explore our advanced PLC programming courses today!