# A Summary of PID Control Algorithms Based on AI-Enabled Embedded Systems

Authors: Yi Zhou
Journal: Security and Communication Networks
 (2022)

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## Abstract

Proportional-integral-derivative (PID) controllers are extensively used in engineering practices for their simple structures, robustness to model errors, and easy operations. At present, there is a great variety of PID controllers. Companies have developed intelligent regulators with functions for automatically tuning PID parameters. For present PID controllers, strategies such as intelligence, self-adaptation, and self-correction are extended to transmission PID. PID controllers and corresponding improved ones are utilized in 90% of industrial control processes. In this paper, PID control algorithms are summarized. This paper focuses on advanced control strategies such as PID control, predictive PID control, adaptive PID control, fuzzy PID control, neural network PID control, expert intelligent PID control, PID control based on genetic algorithms, and PID control based on ant colony algorithms. Besides, these kinds of algorithms are compared, and prospects of PID algorithms are forecast at the end of this paper.


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## Body

## 1. Introduction

Minorsky [1] put forward a method for designing PID controllers based on the output feedback in 1922. By the forties of the last century, PID controllers had been used most widely as regulators in engineering practices. Almost 70 years have passed since the emergence of PID controllers. Because of their simple algorithms, high stability, robustness, reliable working, and convenient regulation, PID controllers have become one of the leading technologies for industrial control. PID regulation is the most technologically mature continuous system which is employed widely. PID control will be the most suitable means if we do not completely understand a system and its controlled objects or we cannot determine system parameters by effective measurement methods. The uses of PID control algorithms are relatively satisfactory in many control fields. Digital PID control algorithms realized by microcomputers, single chip microcomputers, and DSP have been further corrected and improved owing to flexibility of their software systems. There are many types of PID control algorithms, of which the requirements somewhat differ in varying applications.With the development of industries, objects have become more and more complicated. Particularly for large time delay and time-varying and non-linear systems, some parameters are unknown, or change slowly, or have time delay, or have random disturbance, or it is impossible to get relatively accurate digital models. Meanwhile, as people have increasingly more rigorous requirements for quality control, deficiencies of routine PID control have been gradually exposed. Conventional PID control is rarely effective for time-varying objects and non-linear systems. Therefore, routine PID control is considerably limited. In view of this, it has been improved in different aspects, which are mainly introduced as follows. On the one hand, routine PID is structurally improved; on the other hand, fuzzy control, neural network control, and expert control are the most active among existing intelligent controls. Once they are used in combination with routine PID control, they can learn from each other, give play to their respective strengths, and constitute intelligent PID control. This paper primarily sums up development and classification of PID algorithms.


## 2. Basic Principles of PID

### 2.1. Basic Components

PID control is a linear combination of the proportion (P), integral (I), and differential (D) of deviations in a feedback system. These three basic control laws have their respective features [2].


#### 2.1.1. Proportional (P) Control

Proportional controllers only change their signal amplitudes without impacting their phases in controlling changes to input signale (t). Proportional control increases open-loop gains of systems. This part of control is dominant.


#### 2.1.2. Differential (D) Control

Differential controllers determine differential for input signals, and differential reflects the rate of changes to a system, so differential control, a leading mode of predictive regulation, forecasts system variations, increases system damping, and enhances phase margin, thus improving system performances.


#### 2.1.3. Integral (I) Control

Integral, a kind of additive effects, records history of system changes, so integral control manifests effects of histories upon current systems. In general, integral control is not separately adopted but combined with PD control, as shown in Figure1.Figure 1
Schematic diagram of PID.


### 2.2. PID Control Laws

Basic input/output relationships of PID control laws may be conveyed by the differential equation as follows:(1)vt=Kpet+1Ti∫0tetdt+Tddetdt,where e (t) is the input bias of the controller; Kp is the gain in proportional control; Ti1/2 is the integral time constant; and Td is the differential time constant.The corresponding transfer function is as follows:(2)Gs=KP1+1Tis+Tds=KpKi•TiTds+Tis+1s.


## 2.1. Basic Components

PID control is a linear combination of the proportion (P), integral (I), and differential (D) of deviations in a feedback system. These three basic control laws have their respective features [2].


### 2.1.1. Proportional (P) Control

Proportional controllers only change their signal amplitudes without impacting their phases in controlling changes to input signale (t). Proportional control increases open-loop gains of systems. This part of control is dominant.


### 2.1.2. Differential (D) Control

Differential controllers determine differential for input signals, and differential reflects the rate of changes to a system, so differential control, a leading mode of predictive regulation, forecasts system variations, increases system damping, and enhances phase margin, thus improving system performances.


### 2.1.3. Integral (I) Control

Integral, a kind of additive effects, records history of system changes, so integral control manifests effects of histories upon current systems. In general, integral control is not separately adopted but combined with PD control, as shown in Figure1.Figure 1
Schematic diagram of PID.


## 2.1.1. Proportional (P) Control

Proportional controllers only change their signal amplitudes without impacting their phases in controlling changes to input signale (t). Proportional control increases open-loop gains of systems. This part of control is dominant.


## 2.1.2. Differential (D) Control

Differential controllers determine differential for input signals, and differential reflects the rate of changes to a system, so differential control, a leading mode of predictive regulation, forecasts system variations, increases system damping, and enhances phase margin, thus improving system performances.


## 2.1.3. Integral (I) Control

Integral, a kind of additive effects, records history of system changes, so integral control manifests effects of histories upon current systems. In general, integral control is not separately adopted but combined with PD control, as shown in Figure1.Figure 1
Schematic diagram of PID.


## 2.2. PID Control Laws

Basic input/output relationships of PID control laws may be conveyed by the differential equation as follows:(1)vt=Kpet+1Ti∫0tetdt+Tddetdt,where e (t) is the input bias of the controller; Kp is the gain in proportional control; Ti1/2 is the integral time constant; and Td is the differential time constant.The corresponding transfer function is as follows:(2)Gs=KP1+1Tis+Tds=KpKi•TiTds+Tis+1s.


## 3. Classification of PID Control

There are many PID control algorithms and improved PID control algorithms. In this paper, only some classical algorithms are summarized and elaborated.


### 3.1. Predictive PID Control

Smith’s (1958) predictive compensator was one of the first pure plans for lag compensation. His basic thought was to move pure time delay out of the control loop [3].In his algorithm, it is hypothesized that past input variations are the same at each step and equal to current input variations. In practice, these relationships are always not tenable during dynamic responses of systems. The impacts of such approximation can be ignored if systems are subjected to no delay or short delay, but with the increase of delayed steps, it is beyond doubt that the effects upon system robustness will be gradually aggravated. Therefore, Smith’s predictor is integrated into the system to compensate time lag, so that delayed regulated variables are reported to the regulator in advance. Then, the regulator will move ahead of time to eliminate impacts of system time delay, reduce overshoot, improve system stability, speed up regulation, and improve effectiveness of large time delay systems [4, 5].In principle, outputs of a PID controller are fed back to the input terminal of PID through a part for compensation, in order to compensate for controlled objects’ lag. In engineering practices, a Smith predictor is fed back to the PID regulator to overcome pure time delay of the controlled objects, as shown in Figure2.Figure 2
Schematic diagram of Smith’s predictive PID controller.


### 3.2. Adaptive PID Control

In the actual process of industrial control, many controlled mechanisms are highly non-linear and time-varying with pure time delay. Impacted by some factors, process parameters might change, so adaptive PID control is effective for solving these problems. Adaptive PID controllers have strengths of both adaptive control and routine PID controllers. Besides being helpful for automatically identifying explored process parameters, automatically tuning controller parameters, and adapting to changes to controlled process parameters, they are also structurally simple, highly robust, and fairly reliable like conventional PID controllers. Field staff and design engineers are familiar with adaptive PID controllers. With these strengths, adaptive PID controllers have developed into relatively ideal automatic devices for process control [6].They are classified into two major categories. PID controllers, which are based on identification of controlled process parameters, are collectively known as adaptive PID controllers. Their parameter design is dependent upon parameter estimation for controlled process models. The other type of adaptive PID controllers is based on some characteristic parameters of a controlled process such as critical oscillation gains and critical oscillation frequency. They are called non-parametric adaptive PID controllers. Parameters of non-parametric adaptive PID controllers are directly adjusted according to characteristic parameters of processes. Parametric adaptive PID control [2] includes(i)
Adaptive PID pole-placement control: adaptive pole-placement control algorithm was firstly put forward by Wellstead et al. in 1979, which was subsequently improved and deepened by Astrom and Wittenmark [7].(ii)
Adaptive PID control based on cancellation principles: Wittenmark and Astrom firstly put forward parametric adaptive PID control algorithms based on cancellation principles. Further development has been achieved.(iii)
Adaptive PID control based on quadratic performance indices: extensive development has been also achieved in non-parametric adaptive PID control, where parameters are optimized by artificial intelligence.


### 3.3. Fuzzy PID Control

In 1965, Zadeh [8], an expert of cybernetics, developed the fuzzy set theory as a new tool for describing, studying, and dealing with fuzzy phenomena. As to fuzzy control, theories on fuzzy sets are adopted. In particular, it is impossible to get systematic and precise mathematical models in some complicated time-varying and non-linear systems with large time delay. For fuzzy control, precise mathematical models of controlled objects are not needed. Like PID controllers, the control precision of these controllers is high. In addition, the controllers are flexible and adaptive, being highly effective for controlling complicated control systems and high-precision servo systems. They have been quite active in control fields over the past years [9–11].Their basic principles are as follows. Based on traditional PID control algorithms, self-tuning of PID parameters is performed. Laws on fuzzy control are set for adaptive tuning of control parameters by controlling parameter errorsE and variations in errors Ec in order to satisfy the requirements for E and Ec parameters in different control periods. For fuzzy PID control algorithms, functional relationships between KP, KI, Kd and error changes EC are established according to theories on fuzzy sets:(3)KP=f1E,EC,KI=f2E,EC,Kd=f3E,EC.During self-tuning ofKP, KI, and Kd, numerical value of E and EC is determined. The control parameters are self-tuned online in accordance with laws about fuzzy control to satisfy control requirements in different control periods, so that control systems of controlled objects are kept highly dynamic and static. At present, there have been some common fuzzy PID controllers such as fuzzy PI controllers, fuzzy PD controllers, fuzzy PI + D controllers, fuzzy PD + I controllers, fuzzy (P + D) 2 controllers, and fuzzy PID controllers. Figure 3 shows online self-tuning of PID parameters based on laws about fuzzy control.Figure 3
Architecture of fuzzy PID control system.


### 3.4. Neural Network PID Control

The adaptive neurons proposed by Windrow [12] are structurally simple and real time without precisely modelling controlled objects. Based on neural model-free control, scholars [13–15] brought forth neural model-free adaptive PID control methods, identified input signals of neural networks, and designed online correction algorithms in combination with strengths of PID control, thus achieving some outcomes in studying sound dynamic and static properties in control systems.RBF (radial basis function) neural networks [16] are forward networks with three layers, namely, an input layer, a hidden layer, and an output layer. The structure of RBF neural network is shown in Figure 4. Their inputs are processed, weighted, and sent to neurons of the output layer. Only a neuron controls outputs on the output layer.Figure 4
Schematic diagram of neural network.To gain desirable control outcomes in PID control, roles of proportional, integral, and differential in control shall be properly regulated, so that they can coordinate with and restrict each other. These relationships are unnecessarily simple linear combinations, and the optimal relationships can be identified among non-linear combinations with countless variations. By studying system performances, PID control with the optimal combination can be performed, thereby finding out P, I, andD under certain optimal control laws. The architecture of the PID control system based on BP neural networks is shown in Figure 5. The controller is made up of two parts:(i)
Classical PID controller: it exercises closed-loop control over controlled objects and tunes the three parametersKP, KI, and Kd online.(ii)
Neural networks: parameters of PID controllers are regulated according to operation state of systems, in hope of optimizing certain performance indices.Figure 5
Architecture of neural network PID control system.Even though output state of neurons on the output layer corresponds to three adjustable parameters (KP, KI, and Kd) of the PID controller, the steady state of the controller is in line with its parameters under some optimal control laws by independent learning of neural networks and adjustment of weighted coefficients.


### 3.5. PID Control Based on Genetic Algorithms

Genetic algorithms [17], abbreviated as GAs, were efficient, parallel, and globally optimal searching methods firstly put forward by Professor Holland from the Michigan University of the United States in sixties of the last century. They are adaptive and globally optimal probabilistic search algorithms which have formed in simulating genetics and evolution of living creatures under natural environment. Their basic thoughts are as follows. To convert problems to be solved, a group composed of individuals operates a group of genetic operators and repeats the process of generation-evaluation-selection-operation before the optimum solution is searched. The optimization method for PID parameters based on genetic algorithms is helpful for simplifying analytical calculations [18].Genetic algorithms are ways for naturally selecting and solving optimum solutions by simulating natural evolution. For PID control based on genetic algorithms [19–22], actual problems are converted into genetic codes by genetic algorithms at first. In practice, coding methods such as binary coding, floating-point coding, and parametric coding are used. Subsequently, initial populations are generated, and after searching, PID regulation is performed. The schematic diagram is shown in Figure 6.Figure 6
Architecture of PID control system based on genetic algorithms.


### 3.6. PID Control Based on Ant Colony Algorithms

In early 1990s, the Italian scholar Dorigo Macro and some others [23] put forward ant colony algorithms by simulating ants’ behaviors for seeking ways together in the nature. Ant colony algorithms, as new algorithms for simulating evolution, are a type of population-based algorithms for simulating evolution inspired by studies on true ant colonies’ behaviors in the nature and random search algorithms. Ants transfer information by virtue of a material known as pheromone. In the course of their movement, ants leave this substance on the paths they have passed by. Besides, they can perceive this substance during their movement, thus guiding their movement direction. Therefore, collective behaviors of colonies composed by numerous ants reflect a kind of positive feedback. The more the ants walk on certain path, the higher the likelihood for late comers to choose the path. In this case, intensity of pheromone is enhanced. This selection process is referred to as ants’ auto-catalysis, and its principle is a positive feedback mechanism, so the ant system is also called enhanced learning system [24, 25].The optimization of ant colony algorithms for PID control parameters can be schematically described by Figure7. A set of three number sequences, namely, Kp, Ti, Td, is reckoned as a set of three cities. Artificial ants start from S and separately pass by a city of the set. At last, they arrive at point D and conform to criterion functions, thereby finding the optimal path. The path searched by ant colony algorithms for optimizing parameters of PID control [26, 27] reflects that the system has the optimal performance index, which is reflected from node value of three control parameters in an ant colony system. Pheromone is released on nodes that ants have passed, and its concentration changes based on criterion functions rather than path length. The criterion functions shall contain information about nodes that ants have passed and current performance indices of systems.Figure 7
Parameters of PID control based on ant colony algorithms.


### 3.7. Expert Intelligent PID Control

With the development of artificial intelligence, many forms of expert control systems have emerged. Naturally, people have thought of developing PID parameters based on expert experiences [28]. The EXACT expert self-tuning controllers launched by Foxboro (the United States) in 1984 are the most classical ones, and this company applies expert systems in PID controllers.Functions such as self-tuning and self-taught learning realized by the combination of expert control and routine PID control can be used for depicting characteristics of complex systems. Corresponding control strategies can be developed and identified by learning and self-organizing. Scholars have investigated design methods and applications of expert self-tuning PID controllers [29]. For defects of ordinary expert self-tuning PID controllers, they have additionally developed intelligent self-tuning controllers and proposed using staircase signals as system inputs. Thus, the systems need not be frequently started in the process of parametric training. In addition, stair number of given signals is flexibly determined in accordance with actual system changes to satisfy control requirements in some special occasions. Because of their high capacity for self-tuning, structures and parameters of object models vary within a relatively big range.An expert system comprises of two elements:(1)
Knowledge base: it stores knowledge entries about a special field which are summarized in advance and represented in certain format.(2)
Inference mechanism: entries from the knowledge base are used for making inferences, judgments, and decisions by similar methods for solving expert problems.Concerning principles of expert intelligent PID control, measured characteristic parameters are compared with predetermined ones, and their deviations are imported into the expert system, which analyzes requisite corrections of parameters of modulators for eliminating characteristic quantity and imports them into routine PID modulators, so as to correct parameters of the modulators. Meanwhile, the modulators perform operations in accordance with system errors and tuned parameters. Output control signals controlled by system error and generalized objects will be output until characteristic parameters on response curves meet expectations in the controlled process. The schematic diagram of expert intelligent PID is shown in Figure8.Figure 8
Schematic diagram of expert intelligent PID.


### 3.8. Other PID Controls

With the rapid development of computer technologies and intelligent machines, people have begun to make PID control algorithms intelligent for the purpose of further improving control [30–33]. Thus, a series of new improved PID algorithms have been developed, including fuzzy PID controllers, intelligent PID controllers, and neural network PID controllers. With the emergence of these algorithms, PID algorithms have become more functionally complete and played more important roles in controlling industrial processes. Their applications will be further promoted. In addition, there are some other PID control algorithms such as self-tuning PID control, non-linear PID control, and PID control based on the combination of GA and BP neural networks.


## 3.1. Predictive PID Control

Smith’s (1958) predictive compensator was one of the first pure plans for lag compensation. His basic thought was to move pure time delay out of the control loop [3].In his algorithm, it is hypothesized that past input variations are the same at each step and equal to current input variations. In practice, these relationships are always not tenable during dynamic responses of systems. The impacts of such approximation can be ignored if systems are subjected to no delay or short delay, but with the increase of delayed steps, it is beyond doubt that the effects upon system robustness will be gradually aggravated. Therefore, Smith’s predictor is integrated into the system to compensate time lag, so that delayed regulated variables are reported to the regulator in advance. Then, the regulator will move ahead of time to eliminate impacts of system time delay, reduce overshoot, improve system stability, speed up regulation, and improve effectiveness of large time delay systems [4, 5].In principle, outputs of a PID controller are fed back to the input terminal of PID through a part for compensation, in order to compensate for controlled objects’ lag. In engineering practices, a Smith predictor is fed back to the PID regulator to overcome pure time delay of the controlled objects, as shown in Figure2.Figure 2
Schematic diagram of Smith’s predictive PID controller.


## 3.2. Adaptive PID Control

In the actual process of industrial control, many controlled mechanisms are highly non-linear and time-varying with pure time delay. Impacted by some factors, process parameters might change, so adaptive PID control is effective for solving these problems. Adaptive PID controllers have strengths of both adaptive control and routine PID controllers. Besides being helpful for automatically identifying explored process parameters, automatically tuning controller parameters, and adapting to changes to controlled process parameters, they are also structurally simple, highly robust, and fairly reliable like conventional PID controllers. Field staff and design engineers are familiar with adaptive PID controllers. With these strengths, adaptive PID controllers have developed into relatively ideal automatic devices for process control [6].They are classified into two major categories. PID controllers, which are based on identification of controlled process parameters, are collectively known as adaptive PID controllers. Their parameter design is dependent upon parameter estimation for controlled process models. The other type of adaptive PID controllers is based on some characteristic parameters of a controlled process such as critical oscillation gains and critical oscillation frequency. They are called non-parametric adaptive PID controllers. Parameters of non-parametric adaptive PID controllers are directly adjusted according to characteristic parameters of processes. Parametric adaptive PID control [2] includes(i)
Adaptive PID pole-placement control: adaptive pole-placement control algorithm was firstly put forward by Wellstead et al. in 1979, which was subsequently improved and deepened by Astrom and Wittenmark [7].(ii)
Adaptive PID control based on cancellation principles: Wittenmark and Astrom firstly put forward parametric adaptive PID control algorithms based on cancellation principles. Further development has been achieved.(iii)
Adaptive PID control based on quadratic performance indices: extensive development has been also achieved in non-parametric adaptive PID control, where parameters are optimized by artificial intelligence.


## 3.3. Fuzzy PID Control

In 1965, Zadeh [8], an expert of cybernetics, developed the fuzzy set theory as a new tool for describing, studying, and dealing with fuzzy phenomena. As to fuzzy control, theories on fuzzy sets are adopted. In particular, it is impossible to get systematic and precise mathematical models in some complicated time-varying and non-linear systems with large time delay. For fuzzy control, precise mathematical models of controlled objects are not needed. Like PID controllers, the control precision of these controllers is high. In addition, the controllers are flexible and adaptive, being highly effective for controlling complicated control systems and high-precision servo systems. They have been quite active in control fields over the past years [9–11].Their basic principles are as follows. Based on traditional PID control algorithms, self-tuning of PID parameters is performed. Laws on fuzzy control are set for adaptive tuning of control parameters by controlling parameter errorsE and variations in errors Ec in order to satisfy the requirements for E and Ec parameters in different control periods. For fuzzy PID control algorithms, functional relationships between KP, KI, Kd and error changes EC are established according to theories on fuzzy sets:(3)KP=f1E,EC,KI=f2E,EC,Kd=f3E,EC.During self-tuning ofKP, KI, and Kd, numerical value of E and EC is determined. The control parameters are self-tuned online in accordance with laws about fuzzy control to satisfy control requirements in different control periods, so that control systems of controlled objects are kept highly dynamic and static. At present, there have been some common fuzzy PID controllers such as fuzzy PI controllers, fuzzy PD controllers, fuzzy PI + D controllers, fuzzy PD + I controllers, fuzzy (P + D) 2 controllers, and fuzzy PID controllers. Figure 3 shows online self-tuning of PID parameters based on laws about fuzzy control.Figure 3
Architecture of fuzzy PID control system.


## 3.4. Neural Network PID Control

The adaptive neurons proposed by Windrow [12] are structurally simple and real time without precisely modelling controlled objects. Based on neural model-free control, scholars [13–15] brought forth neural model-free adaptive PID control methods, identified input signals of neural networks, and designed online correction algorithms in combination with strengths of PID control, thus achieving some outcomes in studying sound dynamic and static properties in control systems.RBF (radial basis function) neural networks [16] are forward networks with three layers, namely, an input layer, a hidden layer, and an output layer. The structure of RBF neural network is shown in Figure 4. Their inputs are processed, weighted, and sent to neurons of the output layer. Only a neuron controls outputs on the output layer.Figure 4
Schematic diagram of neural network.To gain desirable control outcomes in PID control, roles of proportional, integral, and differential in control shall be properly regulated, so that they can coordinate with and restrict each other. These relationships are unnecessarily simple linear combinations, and the optimal relationships can be identified among non-linear combinations with countless variations. By studying system performances, PID control with the optimal combination can be performed, thereby finding out P, I, andD under certain optimal control laws. The architecture of the PID control system based on BP neural networks is shown in Figure 5. The controller is made up of two parts:(i)
Classical PID controller: it exercises closed-loop control over controlled objects and tunes the three parametersKP, KI, and Kd online.(ii)
Neural networks: parameters of PID controllers are regulated according to operation state of systems, in hope of optimizing certain performance indices.Figure 5
Architecture of neural network PID control system.Even though output state of neurons on the output layer corresponds to three adjustable parameters (KP, KI, and Kd) of the PID controller, the steady state of the controller is in line with its parameters under some optimal control laws by independent learning of neural networks and adjustment of weighted coefficients.


## 3.5. PID Control Based on Genetic Algorithms

Genetic algorithms [17], abbreviated as GAs, were efficient, parallel, and globally optimal searching methods firstly put forward by Professor Holland from the Michigan University of the United States in sixties of the last century. They are adaptive and globally optimal probabilistic search algorithms which have formed in simulating genetics and evolution of living creatures under natural environment. Their basic thoughts are as follows. To convert problems to be solved, a group composed of individuals operates a group of genetic operators and repeats the process of generation-evaluation-selection-operation before the optimum solution is searched. The optimization method for PID parameters based on genetic algorithms is helpful for simplifying analytical calculations [18].Genetic algorithms are ways for naturally selecting and solving optimum solutions by simulating natural evolution. For PID control based on genetic algorithms [19–22], actual problems are converted into genetic codes by genetic algorithms at first. In practice, coding methods such as binary coding, floating-point coding, and parametric coding are used. Subsequently, initial populations are generated, and after searching, PID regulation is performed. The schematic diagram is shown in Figure 6.Figure 6
Architecture of PID control system based on genetic algorithms.


## 3.6. PID Control Based on Ant Colony Algorithms

In early 1990s, the Italian scholar Dorigo Macro and some others [23] put forward ant colony algorithms by simulating ants’ behaviors for seeking ways together in the nature. Ant colony algorithms, as new algorithms for simulating evolution, are a type of population-based algorithms for simulating evolution inspired by studies on true ant colonies’ behaviors in the nature and random search algorithms. Ants transfer information by virtue of a material known as pheromone. In the course of their movement, ants leave this substance on the paths they have passed by. Besides, they can perceive this substance during their movement, thus guiding their movement direction. Therefore, collective behaviors of colonies composed by numerous ants reflect a kind of positive feedback. The more the ants walk on certain path, the higher the likelihood for late comers to choose the path. In this case, intensity of pheromone is enhanced. This selection process is referred to as ants’ auto-catalysis, and its principle is a positive feedback mechanism, so the ant system is also called enhanced learning system [24, 25].The optimization of ant colony algorithms for PID control parameters can be schematically described by Figure7. A set of three number sequences, namely, Kp, Ti, Td, is reckoned as a set of three cities. Artificial ants start from S and separately pass by a city of the set. At last, they arrive at point D and conform to criterion functions, thereby finding the optimal path. The path searched by ant colony algorithms for optimizing parameters of PID control [26, 27] reflects that the system has the optimal performance index, which is reflected from node value of three control parameters in an ant colony system. Pheromone is released on nodes that ants have passed, and its concentration changes based on criterion functions rather than path length. The criterion functions shall contain information about nodes that ants have passed and current performance indices of systems.Figure 7
Parameters of PID control based on ant colony algorithms.


## 3.7. Expert Intelligent PID Control

With the development of artificial intelligence, many forms of expert control systems have emerged. Naturally, people have thought of developing PID parameters based on expert experiences [28]. The EXACT expert self-tuning controllers launched by Foxboro (the United States) in 1984 are the most classical ones, and this company applies expert systems in PID controllers.Functions such as self-tuning and self-taught learning realized by the combination of expert control and routine PID control can be used for depicting characteristics of complex systems. Corresponding control strategies can be developed and identified by learning and self-organizing. Scholars have investigated design methods and applications of expert self-tuning PID controllers [29]. For defects of ordinary expert self-tuning PID controllers, they have additionally developed intelligent self-tuning controllers and proposed using staircase signals as system inputs. Thus, the systems need not be frequently started in the process of parametric training. In addition, stair number of given signals is flexibly determined in accordance with actual system changes to satisfy control requirements in some special occasions. Because of their high capacity for self-tuning, structures and parameters of object models vary within a relatively big range.An expert system comprises of two elements:(1)
Knowledge base: it stores knowledge entries about a special field which are summarized in advance and represented in certain format.(2)
Inference mechanism: entries from the knowledge base are used for making inferences, judgments, and decisions by similar methods for solving expert problems.Concerning principles of expert intelligent PID control, measured characteristic parameters are compared with predetermined ones, and their deviations are imported into the expert system, which analyzes requisite corrections of parameters of modulators for eliminating characteristic quantity and imports them into routine PID modulators, so as to correct parameters of the modulators. Meanwhile, the modulators perform operations in accordance with system errors and tuned parameters. Output control signals controlled by system error and generalized objects will be output until characteristic parameters on response curves meet expectations in the controlled process. The schematic diagram of expert intelligent PID is shown in Figure8.Figure 8
Schematic diagram of expert intelligent PID.


## 3.8. Other PID Controls

With the rapid development of computer technologies and intelligent machines, people have begun to make PID control algorithms intelligent for the purpose of further improving control [30–33]. Thus, a series of new improved PID algorithms have been developed, including fuzzy PID controllers, intelligent PID controllers, and neural network PID controllers. With the emergence of these algorithms, PID algorithms have become more functionally complete and played more important roles in controlling industrial processes. Their applications will be further promoted. In addition, there are some other PID control algorithms such as self-tuning PID control, non-linear PID control, and PID control based on the combination of GA and BP neural networks.


## 4. Comparisons of Key Algorithms for PID Control

In this paper, comparisons and summaries are made based on aforementioned algorithms. As shown in Table1, strengths and weaknesses of algorithms are summarized to fully introduce performances of each algorithm.Table 1
Summarized strengths and weaknesses of PID algorithms.
PID typeTuning techniquesStrengthsWeaknessesPredictive PID controlSmith predictorReduces overshoot, increases system stability, facilitates regulation, and improves large time delay systemsIt is necessary to obtain precise mathematical models of controlled objects for controlled systems during use, and the systems are rather sensitive to modelsAdaptive PID controlAdaptive control systemsAutomatic identification, decision making, and modificationConflicts between control accuracy and parametric estimationFuzzy PID controlCalculates control outputs according to type of input information or checks the list of fuzzy rulesHighly precise, flexible, and adaptiveInapplicable in case of severe system with non-linearity and uncertaintyNeural network PID controlControl parameters are corrected online by BP networksParameters P, I, andD can be determined under certain optimal control laws by self-taught learning of neural networksLow rate of convergencePID control based on genetic algorithmsConverts actual problems into genetic codes for iterationSimplifies optimized analytical calculationsParameter range is so wide that the initial process for seeking optimal solutions is a little purposelessPID control based on ant colony algorithmsCriterion functions by passing pheromoneDetermines the optimal performance indexAlgorithms are rather complex and computing is time consuming. If parameters are improperly determined, lag or local optimization will be caused.Expert intelligent PID controlAdjustments are automatically made according to response characteristics and control requirements of systems, or controller parameters are determined by the companyDescribes characteristics of complex systems and develops corresponding control strategies by self-tuning and self-taught learningEnough knowledge bases and inference mechanisms shall be used as foundations


## 5. Conclusion

PID control, which is widely used, exhibits fairly high capacity when updated. Successfully using PID controllers for controlling complex objects is a main research area. The optimal PID control methods are sought among various models such as fuzzy models, non-parametric predictive models, and expert systems, so that designs of PID controllers can be simplified. Moreover, in order to solve some existing problems in automation and control, PID control is combined with other algorithms to develop more functionally complete controllers. With the rapid development of computer technologies and smart machines, the applications of these controllers will be further promoted. Therefore, it is necessary to further study PID algorithms.



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