Where is a PID controller used in industry?
Production facilities often have many systems that must remain in a certain state and not change under external factors. For example, in a gas pipe, constant pressure must be maintained and not be allowed to fall; a CNC machine needs to adjust the number of spindle revolutions depending on the load on the workpiece, and a robot needs to position its manipulator in a given area with high precision. For all these tasks, a feedback system or a regulator is used.
An object management system can be divided into several components:
· Control object: what needs to be controlled;
· Control device which changes the state of the object depending on the given signal;
· The setpoint is the required parameter value, set by a program or a human worker;
· A regulator calculates the signal value for the control device using a certain algorithm;
· A sensor measures the actual value of the parameter of the object;
· Feedback returns the value of error between the actual value of the parameter and the required one.
Control systems can be different, but the most common ones are: a PID controller and a two-position one.
Two-position regulator
A two-position regulator is a two-position switch that can be in one of two states: on or off.
For example, let’s imagine a water heater that can heat water up to 70°C. The sensor has measured the current water temperature; the regulator has determined that the water temperature is too low and turned on the heating element (tubular electric heater). The temperature increased to 70°C or slightly higher, and the regulator turned off the heating element.
A certain dead band or hysteresis band is introduced to ensure the regulator does not briefly switch on and off. For example, when the water temperature drops to 60°C, the regulator will turn on the heating element again and heat the water to 70°C.
In this case, the exact temperature is not as important to the end-user as the overall availability of hot water. If we need our water at an exact temperature, then another type of controller is used, for example, a PID controller.
PID controller
A proportional-integral-derivative (PID) controller is a device that generates a signal to control the state of an object and its parameters based on feedback. The regulator can be a separate hardware device or implemented in software on a PLC or computer. The PID controller enables to smoothly and accurately bring the control object to the desired state with the required parameters. For example, it enables to maintain a constant engine speed, maintain the temperature of a soldering iron tip at a given level, and print a 3D model with high accuracy on a 3D printer.
From the very name of the PID controller, it is obvious that it consists of three components:
· Proportional
· Integral
· Derivative
If any one of these components is not used, the regulator will be called: Proportional-Integral (PI), Proportional-Derivative (PD), Proportional (P), etc.
The PID controller formula is an expression with various coefficients
Where Kp, Ki, Kd are the gains of the proportional, integral and derivative components of the controller, respectively. The function is time-dependent since we need to periodically check the state of the object and quickly adjust it. Equal time intervals between calculations of the formula are called sampling times.
All that is needed for the PID controller to work is to select the correct coefficients, and the controller itself will perform the calculations using the formula.
The clear influence of the coefficients on the transient response of the system can be seen in the animated gif in Wikipedia.
Proportional component
The proportional component in the regulator is the output signal at the current moment in time, which reduces the control error between the actual value of the parameter and the required value. It is equal to the control error multiplied by a constant factor.
The level of this signal depends on the magnitude of the error. Hence, the value of the output signal decreases as it approaches the required value. Around the required value, the output signal is so weak that its level does not allow the error to be reduced to zero, and a static error occurs. If the coefficient of the proportional component is set large enough, then the object can go into self-oscillation mode, and if it is set small, then the parameter will tend to the desired value over a long time.
In order to remove the static error, the integral component is added.
Integral component
The integral component outputs a signal proportional to the accumulated error multiplied by a constant factor. Even if the error between the actual value of the parameter and the required one is small, its value accumulates over time, which makes it possible to generate a signal to compensate for the static error.
Derivative component
The derivative component responds to the rate of change of the parameter, meaning it is the time derivative of the error multiplied by a constant factor. The greater the change, the greater the value of the output signal. The derivative component enables to compensate for sudden changes in the system, prevent severe overshoot and reduce system buildup.
How to configure a PID controller
There are many ways of configuring a PID controller, from accurate calculations using the formula to the experimental selection of coefficients. Some manufacturers even supply their PID controllers with automatic parameter configuration when the PID controller itself chooses the optimal coefficients based on the response of the actual system to the control action.
The most popular method of configuring a PID controller is the Ziegler-Nichols method, which enables to experimentally select coefficients. The advantage of the method is its simplicity, and the main disadvantage is the small margin of stability of the system. The method offers two options for setting the controller parameters.
Ziegler-Nichols method, first option
It is based on the frequency characteristics of the control object. The configuring procedure begins with an experimental test of the system with only a P-controller. The rest of the coefficients must be set to 0 so that they do not affect the system. The coefficient of the P-controller should be experimentally increased until constant amplitude oscillations are established at the output of the system, and the system is at the stability boundary. Then the value of this coefficient k and the period T of steady oscillations are used to calculate the remaining coefficients.
Depending on the type of controller, the values of the coefficients are calculated using the following formulas:
The disadvantage of this option is the need to bring the system to the stability boundary, which is not allowed for some control objects.
Ziegler-Nichols method, second option
The method uses the object’s response to a single change. This characteristic of the object is usually called the acceleration curve. Control objects that have an aperiodic acceleration curve, as shown in the figure below, are represented in the form of a series connection, aperiodic and lagging links. The response of the control object to the step action will look like this:
In this case, the transfer function of the object is as follows:
where k is the transmission coefficient, T is the time constant, t is the delay time.
The values of the controller parameters are calculated directly from the values of the parameters k, T, t. Formulas for calculating the controller parameters are shown in the table below. The method gives satisfactory results if 0.15 < t / T < 0.6.
The Ziegler-Nichols method does not always provide the required quality of the control process; therefore, additional manual adjustment of the controller parameters is often required. Despite this, the Ziegler-Nichols method is popular and is often used to configure controllers.
Manual configuration
After configuring a PID controller using the Ziegler-Nichols method, the controller parameters may not be optimal and require manual adjustment. Coefficient adjustment rules may be used to this end.
Increasing the coefficients provides the following effect:
Conclusion
The field of automated control is constantly evolving. New types of controllers and methods of adjusting them are emerging, including with the use of Artificial Intelligence and neural networks. Modern configuring tools enable to analyze valve wear, assess robustness, and automatically generate reports with control loop parameters and much more. This ultimately simplifies the work of process control engineers.
