Editor's Note
PID stands for Proportional-Integral-Calculator, which you can find in detail in an automatic control course! In temperature control, direct action means heating, and reverse action means cooling.
Introduction to PID Control
Currently, the level of industrial automation has become an important indicator of the modernization level of various industries. Simultaneously, the development of control theory has gone through three stages: classical control theory, modern control theory, and intelligent control theory. A typical example of intelligent control is the fuzzy fully automatic washing machine. Automatic control systems can be divided into open-loop control systems and closed-loop control systems. A control system includes a controller, sensors, transmitters, actuators, and input/output interfaces. The controller's output is applied to the controlled system through the output interface and actuators; the controlled variable of the control system is sent to the controller through the sensors and transmitters via the input interface. Different control systems use different sensors, transmitters, and actuators. For example, a pressure control system uses a pressure sensor. An electric heating control system uses a temperature sensor. Currently, PID control and its controllers or intelligent PID controllers (instruments) are widely available, and these products have been widely used in engineering practice. Various PID controller products are available, and major companies have developed intelligent regulators with PID parameter self-tuning functions. The automatic adjustment of PID controller parameters is achieved through intelligent adjustment or self-calibration, and adaptive algorithms. There are pressure, temperature, flow, and level controllers that utilize PID control, programmable logic controllers (PLCs) that can implement PID control, and PC systems that can implement PID control, etc.
1. Open-loop control system
An open-loop control system is one in which the output of the controlled object (the controlled variable) has no effect on the output of the controller. In such a control system, there is no need to feed the controlled variable back to form any closed loop.
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2. Closed-loop control system
A closed-loop control system is characterized by the fact that the output of the controlled object (the controlled variable) is fed back to affect the controller's output, forming one or more closed loops. Closed-loop control systems can have positive or negative feedback. If the feedback signal is opposite to the system's setpoint signal, it is called negative feedback; if the polarities are the same, it is called positive feedback. Generally, closed-loop control systems use negative feedback, hence the name negative feedback control system. There are many examples of closed-loop control systems. For instance, the human body is a closed-loop control system with negative feedback; the eyes act as sensors, providing feedback, and the human system can make various correct actions through continuous correction. Without eyes, there would be no feedback loop, resulting in an open-loop control system. Another example is a fully automatic washing machine that continuously checks whether the clothes are clean and automatically cuts off the power after washing; this is a closed-loop control system.
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3. Step response
A step response is the system's output when a step input is applied. Steady-state error is the difference between the system's expected output and its actual output after the system's response reaches a steady state. The performance of a control system can be described by three words: stable, accurate, and fast. Stable refers to the system's stability; for a system to function properly, it must first be stable, and from the perspective of a step response, it should converge. Accurate refers to the control system's precision and precision, usually described by steady-state error, which represents the difference between the system's steady-state output value and its expected value. Fast refers to the speed of the control system's response, usually quantitatively described by rise time.
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4. Principles and characteristics of PID control
In engineering practice, the most widely used regulator control law is proportional-integral-derivative control, abbreviated as PID control, also known as PID regulation. The PID controller has been around for nearly 70 years, and its simple structure, good stability, reliable operation, and convenient adjustment have made it one of the main technologies in industrial control. When the structure and parameters of the controlled object cannot be fully understood, or when an accurate mathematical model is unavailable, and other control theory techniques are difficult to apply, the structure and parameters of the system controller must be determined based on experience and on-site debugging. In this case, PID control technology is the most convenient. That is, when we do not fully understand a system and the controlled object, or cannot obtain system parameters through effective measurement methods, PID control technology is most suitable. In practice, PI and PD control also exist. The PID controller calculates the control quantity based on the system error using proportional, integral, and derivative calculations.
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Proportional (P) control
Proportional control is the simplest control method. Its controller output is proportional to the input error signal. When only proportional control is used, the system output exhibits a steady-state error.
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Integral (I) control
In integral control, the controller's output is proportional to the integral of the input error signal. For an automatic control system, if a steady-state error exists after reaching steady state, the control system is said to have a steady-state error, or simply a system with error (System with Steady-State Error). To eliminate steady-state error, an "integral term" must be introduced into the controller. The integral term depends on the integral of the error over time; as time increases, the integral term increases. Thus, even if the error is small, the integral term will increase with time, driving the controller's output to increase and further reduce the steady-state error until it equals zero. Therefore, a proportional-integral (PI) controller can ensure that the system has no steady-state error after reaching steady state.
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Differential (D) control
In differential control, the controller's output is proportional to the derivative of the input error signal (i.e., the rate of change of the error). Automatic control systems may experience oscillations or even instability during error correction. This is because of components with significant inertia or delay, which suppress error but whose changes always lag behind the error's change. The solution is to make the error-suppressing effect "lead," meaning that when the error approaches zero, the error-suppressing effect should be zero. This means that simply introducing a proportional term in the controller is often insufficient; the proportional term only amplifies the error's amplitude. What's needed is a derivative term, which predicts the trend of error change. Thus, a controller with a proportional-derivative (PD) term can make the error-suppressing control effect equal to zero, or even negative, in advance, thus avoiding severe overshoot of the controlled variable. Therefore, for controlled objects with significant inertia or delay, a PD controller can improve the system's dynamic characteristics during adjustment.
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5. PID controller parameter tuning
PID controller parameter tuning is a core aspect of control system design. It involves determining the proportional gain, integral time, and derivative time of the PID controller based on the characteristics of the controlled process. There are many methods for PID controller parameter tuning, which can be broadly categorized into two main types:
One method is the theoretical calculation tuning method. This method mainly relies on the mathematical model of the system to determine the controller parameters through theoretical calculations. The calculated data obtained by this method may not be directly usable and must be adjusted and modified based on actual engineering practice.
Secondly, there are engineering tuning methods. These primarily rely on engineering experience and are performed directly in the testing of the control system. They are simple, easy to master, and widely used in practical engineering. Engineering tuning methods for PID controller parameters mainly include the critical proportional method, the response curve method, and the decay method. Each of these three methods has its own characteristics, but they all share the common feature of tuning the controller parameters through experimentation and then according to empirical formulas. However, regardless of the method used, the controller parameters obtained still require final adjustment and refinement during actual operation.
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The critical proportional method is generally used nowadays. The steps for tuning PID controller parameters using this method are as follows:
(1) First, pre-select a sufficiently short sampling period for the system to operate;
(2) Add only the proportional control loop until the system exhibits critical oscillation in response to a step input. Record the proportional gain and the critical oscillation period at this point.
(3) The parameters of the PID controller are calculated by formula under a certain degree of control.
Setting PID parameters relies on experience and familiarity with the process, referring to the curve of measured value tracking and set value, thereby adjusting the magnitude of P, I, and D.
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For engineering tuning of PID controller parameters, the following empirical data on PID parameters in various control systems can be used as a reference:
Temperature T: P = 20~60%, T = 180~600s, D = 3-180s
Pressure P: P = 30~70%, T = 24~180s,
Liquid level L: P = 20~80%, T = 60~300s
Flow rate L:P = 40~100%, T = 6~60s.
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Commonly used mnemonics in the book:
Find the optimal parameters by checking in ascending order.
First the proportion, then the integral, and finally the differential.
The curve oscillates very frequently, so the scale plate needs to be enlarged.
The curve floats around the large bay, the scale dial is turned smaller.
The curve deviates from its normal response, resulting in a slower integral time.
The curve has a long fluctuation period, and the integration time is further extended.
The curve oscillates at a fast frequency, so we need to reduce the derivative first.
Larger momentum results in slower fluctuations. The differential time should be increased.
The ideal curve has two waves, with the first wave higher than the second in a 4:1 ratio.
With careful observation, thorough investigation, and comprehensive analysis, the quality of regulation will not be low.
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This section introduces an empirical method. This method is essentially a trial-and-error approach, a proven method summarized from production practice and widely applied on-site.
The basic procedure of this method is to first determine a set of regulator parameters based on operational experience, and then put the system into closed-loop operation. Next, a step disturbance is artificially introduced (such as changing the regulator's setpoint), and the step response curve of the controlled variable or the regulator output is observed. If the control quality is deemed unsatisfactory, the regulator parameters are changed according to the influence of each tuning parameter on the control process. This process is repeated until satisfactory results are achieved.
The empirical method is simple and reliable, but it requires a certain amount of field operating experience and is prone to subjectivity and bias during tuning. When using a PID controller, there are multiple tuning parameters, and the number of repeated trials increases, making it difficult to obtain the optimal tuning parameters.
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The following uses a PID controller as an example to illustrate the tuning steps using the empirical method:
(1) Set the integral coefficient S0 of the regulator parameter to 0 and the actual derivative coefficient k to 0, put the control system into closed-loop operation, change the proportional coefficient S1 from small to large, make the disturbance signal make a step change, observe the control process, and continue until a satisfactory control process is obtained.
(2) Take the proportional coefficient S1 as the current value multiplied by 0.83, and increase the integral coefficient S0 from small to large. Similarly, make the disturbance signal change stepwise until a satisfactory control process is obtained.
(3) Keep the integral coefficient S0 unchanged, change the proportional coefficient S1, and observe whether the control process improves. If it improves, continue to adjust until you are satisfied. Otherwise, increase the original proportional coefficient S1 and then adjust the integral coefficient S0 to try to improve the control process. Repeat this trial and error until you find a satisfactory proportional coefficient S1 and integral coefficient S0.
(4) Introduce appropriate actual differential coefficients k and actual differential time TD. At this time, the proportional coefficient S1 and integral coefficient S0 can be appropriately increased. Similar to the previous steps, the differential time tuning also needs to be repeatedly adjusted until the control process is satisfactory.
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Note: The PID controller used in the simulation system is different from the traditional industrial PID controller. The parameters are isolated from each other and do not affect each other, making it very convenient to observe the regulation law.
PID parameters are determined based on the inertia of the controlled object. For large inertia, such as temperature control in a large drying oven, P is typically above 10, I = 3-10, and D = around 1. For small inertia, such as a small motor driving a water pump in closed-loop pressure control, only PI control is generally used. P = 1-10, I = 0.1-1, and D = 0. These parameters need to be corrected during on-site commissioning.
I'll provide an incremental PID for your reference.
△U(k)=Ae(k)-Be(k-1)+Ce(k-2)
A = Kp(1 + T/Ti + Td/T)
B=Kp(1+2Td/T)
C=KpTd/T
T is the sampling period, Td is the differential time, and Ti is the integral time.
The algorithm above can be used to construct your own PID algorithm.
U(K) = U
