Review of PID control design and tuning methods

Owing to the fact that PID controllers have a simple and fixed form and good robustness, they are widely used in industrial production. However, the complexity and non-linearity of control systems affect the use of a traditional PID controller. Therefore, how to design a stable PID controller that does not require an accurate mathematical model or how to adjust the parameters of the PID controller have become hot topics of research today and in the future. PID controllers are outlined in this study, such as position PID controllers, incremental PID controllers, differential prior PID controllers, fuzzy PID controllers. Besides, two different PID controller rectification methods are described: the model-based parameter self-tuning method and the rule-based self-tuning method. What is more, this paper also compares these PID controllers, and clarifies the advantages and disadvantages of each PID controller. And at the end, it predicts the prospect of PID controllers and PID controller rectification methods. PID has a great development prospect, and its future research direction is not only to study new algorithms, but also needs to study how to combine the excellent existing PID controllers to make up for their shortcomings.


Introduction
PID controllers have been around for a century, and nowadays they are widely used in various control fields.However, whether PID controllers can have a satisfactory effect depends on the design of the PID controller on the one hand, and the PID controller rectification method on the other.
With the development of industry, today's production processes have become increasingly complex, with non-linear, time-varying uncertainties.Then, it is hard for people to get an accurate mathematical model.Therefore, traditional PID controller are no longer perfectly suited to the needs.Faced with this problem, more and more new PID controllers are designed, such as the position PID controller, the incremental PID controller, the differential PID controller, the fuzzy PID controller, and so on.Besides, the PID controller rectification method is also essential.This is because it can help the system continue working well, even if the model parameters change during the process.According to the working mechanism, it can be divided into model-based parameter self-tuning and rule-based self-tuning method.This paper focuses on the development and classification of PID controllers and the classification of PID controller rectification methods.By analyzing the problems and challenges faced by PID controllers, the development of PID controllers is prospective.

History of PID development
The development of PID control can be divided into two stages [1].In the late 1930s, with the addition of differential control, the PID algorithm was formally formed, and it was also a watershed in the development of two stages of PID control.The first stage is the invention stage   [2].In 1922, Minorsky used the analysis of errors to create an autonomous ship steering system for the U.S. Navy., which represents the first method for designing a PID controller.After that, PID controllers were used in the analysis and design of automatic control systems and industrial processes.PID goes into practice.After 1940, there was the second stage (the innovation stage) [2].In this stage, PID controller has become a strong robustness, easy to adjust, reliable and stable controller.Until now, PID controllers are still widely used nowadays and are often used in automation, industrial control and other related fields.With the emergence of various technologies, algorithms, the popularity of PID controllers has not been weakened.On the contrary, more and more accurate and stable PID controllers have emerged.What is more, the development of certain PID control techniques requires more accurate PID control, thus stimulating the development of PID controller design and parameter tuning techniques [2].

PID controller design method
A PID controller consists of three components: P(proportional), I(Integral) and D(Derivative).Various combinations of these can be used to generate various PID controllers with different functions.
PID controllers have a wide range of applications, and for different control objects, the requirements for PID control are often different.For example, the PI controller mainly focuses on rapidity, reducing or eliminating the static difference.However, PD controllers offer speed and stability.In general, the design of a PID controller needs to meet the following requirements [3]: The PID controller meets performance metrics; the PID controller is based on a process that is known or available only; The PID controller meets computational capacity constraints, and the resources required for the design are accessible.

Position PID controller design method
PID control is a second-order linear controller.The original metric of the PID algorithm formula is: In the above equation, u(t) is the output of the control system, and e(t) is the input of the control system, which is the error between the target value and the current value.K p is the scale factor of the control system.T i is the integration time of the control system and T d is the differentiation time of the control system.
The equation above can also be written in difference equation form: T means the sampling period in this equation.e(k) means the deviation signal obtained in the k-th sampling period.e(k − 1) means the deviation signal obtained in the (k − 1) − th sampling period.
The position PID control algorithm is suitable for actuators without integral elements, where the actuator position corresponds to the input signal in a one-to-one relationship.The controller calculates the control variable output after the k − th sampling based on the deviation e(k) between the k − th sampling result of the controlled variable and the set value [4].
The position PID controller has some disadvantages: The sampled output quantity is associated with any previous state and is not a separate control quantity.Besides, the calculation is large and requires the accumulation of e(k).What is more, if an accidental calculation error occurs, it will cause the actuator to make a large movement, which is prone to danger.

Incremental PID controller design method
In the position PID controller, each output and control deviation is related to the entire change process in the past, so that the cumulative effect of deviations can easily produce large cumulative deviations, making the control system appear undesirable overshoot phenomenon [5].Besides, some actuators do not need the absolute values of control variables but need the increment, and incremental PID controllers meet this design requirement.For example, In valve control, only the change part of the valve opening needs to be output.
Therefore, Incremental PID controller is needed.The equation is shown below: Then, equation 3 is subtracted from equation 2, and the equation for incremental PID controller can be obtained: (5) According to the equations above, only the deviations e(k) , e(k − 1) , e(k − 2) within three sampling periods need to be guided to calculate the increment of the control variable ∆u within this sampling period.
Although the incremental PID control algorithm is only a small change or improvement in the algorithm, it has lots of advantages.For example, there is no need to add up the errors in the equation; the increment is only related to the deviations, which is easy to obtain a better control effect by weighing.Besides, since the computer output is incremented, the impact on the output in case of a malfunction is relatively small and can be eliminated if necessary by using logic judgment [6].
However, an incremental PID controller also has some disadvantages: it has a large integral truncation effect and steady-state error.Some controlled objects are not so good with an incremental PID controller.

Differential prior PID controller design method
The traditional PID controller generally has the phenomenon of integral saturation, which leads to the system runaway and excessive overshoot.In response to this problem, a differential prior PID algorithm has emerged, which essentially advances the differential operation.The essence of a differential prior algorithm is to differentiate the output quantity in advance.
The transfer function of the differential part is: In this equation, γ < 1.
1 γT D S+1 is equal to Low-pass filter.Using the difference form, equation 7 can be transformed into the equation below: In order to better compare differential prior PID controller and traditional PID controller.The equation of differential part of a traditional PID controller is shown below: Compare equation 8 and equation 9, differential prior PID controller only differentiate h(k), which is the output value.However, both the output value h(k) and the set value r(k) are differentiated.A differential advance PID controller performs differential operation on the output in advance but does not differentiate the set value, so that the system can overcome overshoot better and avoid system oscillation caused by the rise and fall of the set value.However, the introduction of simple differential advance will also make the change of controlled quantity moderate, and the anti-interference ability of the system also needs to be further improved [7].

Fuzzy PID controller design method
Zadeh, a cybernetics expert, created the fuzzy set theory in 1965 as a novel method for describing, researching, and understanding fuzzy phenomena [8].Although traditional PID controllers are effective for simple linear systems, they are often not effective for nonlinear systems, higher-order systems, timelagged systems, time-varying systems, and so on.For instance, it is challenging to develop a precise mathematical model because real-world industrial production processes frequently involve nonlinear, time-varying uncertainty.Therefore, in this case, traditional PID cannot achieve the desired effect.For a fuzzy PID controller, it is an established engineering control technique that doesn't overly complicate the control structure [9], which stems from the fact that there is no need to establish precise mathematical models.Thus, it can solve lots of problems in today's industrial field or automation control field.
The following are the fundamentals of a fuzzy PID controller.The quantization function, which is often an interval of numbers symmetric with regard to 0, first projects the input amount to a particular numerical level.Because it directly influences the calculation's accuracy, the precise interval to be projected depends on the current condition.In the fuzzy algorithm mentioned above, fuzzification is a crucial step.Following the identification of the fuzzy subsets corresponding to each linguistic variable, the set to which the input belongs may be determined based on the quantified results, and the appropriate affiliation degree can then be calculated.There are numerous methods for calculating the affiliation degree, but the most popular ones are the trapezoidal or triangular affiliation functions.The fundamental component of fuzzy control is inference decision, which simulates human reasoning and decisionmaking by using data from the knowledge base and fuzzy operation methods.Under specific input conditions, the corresponding control rules are activated to produce the desired output for fuzzy control.Finally, after fuzzy inference, a series of fuzzy expressions will be obtained, at which point a defuzzification operation is performed to obtain the data.The architecture of a fuzzy PID controller is shown above.According to this figure, a fuzzy inference system (FIS) is used to tune all three PID parameters [10].The input of fuzzy inference is the error and the rate of change of the error.The output of fuzzy inference is three parameters: K p , K i , K d .The values of inputs and outputs are not precise, they are fuzzy.What is more, they are related to the experience of manually tuning the PID controller.
Fuzzy PID controllers are widely used today, and fuzzy control techniques are applied to a variety of different PID controllers, such as fuzzy PI controllers, fuzzy PD controllers, fuzzy PI + traditional D controllers, and so on.

PID controller rectification method
In the early days, the PID parameters were basically adjusted manually, according to the engineer's test and adjustment of the actuator, and then the parameters were obtained from the adjustment formula.However, in modern industrial production, some products often include the use of dozens or hundreds of PID controllers, and the manual adjustment of PID parameters is obviously inappropriate.

Model-based parameter self-tuning method
Model-based self-tuning methods require model identification.The parametric model identification method first assumes the process as a model structure and then determines the parameters of the model.If the model structure cannot be fully determined, some structural identification methods are used to first determine the model structure (e.g., the order of the model).The methods used for parametric model identification are least squares, gradient, and maximum likelihood.In the process of deriving the formula for adjusting the parameters of the internal mode control PID controller, the time lag needs to be approximated if the target is a time lag object [11].
PID algorithms are the most widely utilized feedback control techniques in industry today because to the well-known loop rectification method developed by John Ziegler and Nathaniel Nichols.Today, many people still make use of the 1942 publication that introduced the Z-N rectification method.Two steps make up the Z-N method: First the stability limit is established, followed by the construction of a closed-loop control loop.Second, the controller parameters are calculated according to Equation.

Rule-based self-tuning method
The rule-based self-tuning method does not require the acquisition of a process model, and the tuning rules are similar to manual tuning by an experienced operator.The rule-based self-tuning process, like the model-based approach, uses information such as step response, set point response, or load disturbance to observe the characteristics of the controlled process and adjust the controller parameters based on the rules if the controlled quantity deviates from the set point [2,11].
In order to obtain a rule-based self-tuning process, the response characteristics need to be quantified, usually using quantities such as overshoot, decay ratio, etc. that describe the stability of the control system.Due to the difficulty of determining the corresponding quantities, rule-based self-tuning methods are more suitable for the field of continuous adaptive control.

Conclusion
Through the above analysis, it is fair to conclude that no matter which PID controller design method or PID controller rectification method is used, they are not foolproof.Therefore, it should be considered to complement various PID design methods to give full play to their strengths and make up for their weaknesses.For example, a fuzzy PID controller can be combined with a position PID controller or an incremental PID controller.What is more, the ideas used in the design of PID controllers can also be used in the PID controller rectification method, such as the fuzzy algorithm.This is because a good selftuning function is very important for the PID controller, and can help it be more stable.In this paper, I do not present in detail how to use the idea of designing a PID controller for the rectification method of a PID controller, but only present an idea.In the future, the author will tend to research in the fields of automation, embedded, etc., and hope to research a more excellent PID controller.

Figure 1 .
Figure 1.Structure of differential forward PID control [7].According to the figure 1 above, it is fair to conclude that E(S) = R(S) − U D (S).The transfer function of the proportional and integral part is: U(S) E(S) = K p �1 + 1 T I S + 1 � (6)

Figure 2 .
Figure 2. The architecture of fuzzy PID controller.