Research on Improved Intelligent Control Processes Based on Three Kinds of Artiﬁcial Intelligence

: Autotuning and online tuning of control parameters in control processes (OTP) are widely used in practice, such as in chemical production and industrial control processes. Better performance (such as dynamic speed and steady-state error) and less repeated manual-tuning workloads in bad environments for engineers are expected. The main works are as follows: Firstly, a change ratio for expert system and fuzzy-reasoning-based OTP methods is proposed. Secondly, a wavelet neural-network-based OTP method is proposed. Thirdly, comparative simulations are implemented in order to verify the performance. Finally, the stability of the proposed methods is analyzed based on the theory of stability. Results and e ﬀ ects are as follows: Firstly, the proposed control parameters of online tuning methods of artiﬁcial-intelligence-based classical control (AI-CC) systems had better performance, such as faster speed and smaller error. Secondly, stability was veriﬁed theoretically, so the proposed method could be applied with a guarantee. Thirdly, a lot of repeated and unsafe manual-based tuning work for engineers can be replaced by AI-CC systems. Finally, an upgrade solution AI-CC, with low cost, is provided for a large number of existing classical control systems.


Introduction
The proportion integration differentiation (PID) control method is widely used in practical control processes. Manual-based parameter-tuning methods of control systems and processes are common in industrial applications [1]. However, it requires experienced engineers to monitor the whole system in real-time and adjust the parameters in a timely manner [2]. Autotuning and online tuning of PID parameters play an important role in applications of PID control. Using autotuning and online tuning techniques, control parameters can be tuned automatically.
The motivation of this study is as follows: (1) Improve the online tuning methods in order to obtain better performance, such as faster control speed and smaller steady-state errors. (2) Analyze and prove the stability of the proposed methods theoretically. (3) Find a low-cost upgrade solution for the existing practices and applications of the PID control process.
The existing autotuning and online tuning methods of control parameters are as follows: (1) Traditional autotuning methods. Firstly, the model-identification-based autotuning control method (MIC) is well known and widely recognized and applied. Ziegler and Nichols presented two classical methods for determining the values of proportional gain, integral time, and derivative time based on the transient response characteristics of a given plant or system. Secondly, the relay autotuner-based autotuning control method (RAC) is an effective method. The relay autotuner provides a simple way to find PID controller parameters without carefully choosing the sampling rate from prior knowledge of the process [3]. An asymmetric relay autotuner, with features such as a startup procedure and adaptive relay amplitudes, has been proposed by Astrom [4]. Soltesz [5] proposed an automatic tuning strategy based on an experiment, followed by simultaneous identification of LTI model parameters and an estimate of their error covariance.
(2) Advanced online tuning methods: Firstly, adaptive control (ADC) can continuously identify model parameters and adjust control parameters to adapt to the changes in the work environment [6]. There are two types of existing adaptive PID control methods: direct and indirect adaptive PID control. Secondly, gain scheduling control (GSC) can solve nonlinear problems well because the gain scheduling controller is formed by interpolation among a set of linear controllers. Gain scheduling control is a simple extension of linear design methods and it is mature for industrial applications. Thirdly, predictive control (PDC) is based on a certain model. PDC uses past input and output to predict output at a certain period of time in the future [7]. Then, PDC minimizes the result of the quadratic objective function with control constraints and prediction errors; the optimal control law of the current and future cycles is obtained. Fourthly, artificial intelligent control (AIC) methods are proposed based on AI. The control variable of AIC, which is the output of the AI controller and the input of the controlled object, is directly calculated by AI methods. The existing AIC methods are as follows: expert control, fuzzy control, and neural network control. AIC methods have strong fault tolerance [8], which can be used for nonlinear and complex control.
(3) AI-based improved classical control (AI-CC) with online-tuning. Expert system, fuzzy calculation, and neural network are typical AI methods that are applied to expert-system-based PID (E-PID), fuzzy-calculation-based PID (F-PID), and neural-network-based PID (NN-PID) methods. In AI-CC systems, AI is an intelligent module to adjust the parameters of a classical controller. The input of the AI module is the same as the input of the PID controller, and the outputs of the AI module are the adjusted K p , K i , K d . The output of the AI module is the input of the CC module.
Differences among the above three methods are as follows: (1) MIC and RAC are autotuning methods, which can automatically adjust the parameters more accurately; the adjustment processes of MIC and RAC are not production processes. GSC, AIC, and AI-CC methods are online-tuning methods, where the adjustment processes can directly be the production processes. (2) MIC, RAC, GSC, and AI-CC are based on classical PID methods, where the tuning process is the optimization process of K p , K i , K d . ADC and PDC have many forms of control models, some of which are not based on adjusting PID parameters. (3) RAC has two steps to adjust PID parameters, while AI-CC has only one step to adjust PID parameters. The production process is the final process of both methods of RAC and AI-CC, while RAC has a unique relay-based autotuning process.

Related Work
The current state-of-the-art research has proven that artificial intelligence (AI) can do a good job of online control work, namely, "Alpha Star in 2019" and "Alpha Go in 2016". Google researchers have published a research report on 49 game control applications with AI methods in Nature [9]. The report shows that (1) humans can be defeated by AI systems in a game actor control process, (2) operations of game control (actor control) can be learned by AI, and (3) a better job can be done than humans in playing many games. Since 2012, Google, Apple, and Tesla have innovated a lot of intelligent control systems [10], such as Siri, Cortana [11], and driverless (car-driving without people). The above research shows that (1) the ability of the latest AI methods has been greatly improved, (2) deep learning [12,13] and reinforcement learning [14] methods have been applied in many different control systems; (3) parameters of the controlled system can be optimized in real-time and online by AI methods, and (4) AI control methods have been widely used in both research domains and business applications. Definition 2. Simulation cycle (SC). 1 SC is an independent control process, which is from the start time of the control process at time = 1 to the end time (e.g., max simulation time _ = 3000) of the control process (before the end time, the system will usually reach steady-state and remain stable). 1 SC is composed of AC.
The model of a classical PID controller can be defined as Equation (1): The model of an AI-CC controller can be defined as Equation (2): ( ), ( ), ( ) are the control parameters that will be online-tuned in each adjustment cycle by the AI-CC methods proposed in this study. Definition 1. Adjustment cycle (AC). 1 AC is only one control process at one fixed simulation time (e.g., t = 3).
In each AC, system errors error(t) and d error (t), updated parameters kp(t), ki(t), kd(t), control amount u(t), and updated system output yout(k) will be updated once.
Definition 2. Simulation cycle (SC). 1 SC is an independent control process, which is from the start time of the control process at time t = 1 to the end time (e.g., max simulation time max_t = 3000) of the control process (before the end time, the system will usually reach steady-state and remain stable). 1 SC is composed of n AC.
The model of a classical PID controller can be defined as Equation (1): The model of an AI-CC controller can be defined as Equation (2): K p (t), K i (t), and K d (t) are the control parameters that will be online-tuned in each adjustment cycle by the AI-CC methods proposed in this study. In each adjustment cycle at time t, the output of the controller is u(t) = u(t − 1) + ∆u(t). The updated u(t) is calculated according to the incremental PID method, which is expressed as Equation (3): The output of the control system is y(t), which can be calculated according to the method of digital PID for discrete systems. Firstly, the transfer function of the controlled object is discretized to a(i) and b( j). Secondly, y(t) is calculated as Equation (4).
The pseudocode for the PID control method can be presented as Algorithm 1:
End while 10. End

E-PID and F-PID Models (Existing Methods)
E-PID and F-PID are typical systems for AI-CC parameter online-tuning with the intelligence of expert and fuzzy rules. The structure of the E-PID and F-PID control systems are shown in Figure 2. In each adjustment cycle at time , the output of the controller is ( ) = ( − 1) + ∆ ( ). The updated ( ) is calculated according to the incremental PID method, which is expressed as Equation (3): The output of the control system is ( ), which can be calculated according to the method of digital PID for discrete systems. Firstly, the transfer function of the controlled object is discretized to ( ) and ( ). Secondly, ( ) is calculated as Equation (4).
The pseudocode for the PID control method can be presented as Algorithm 1:

E-PID and F-PID Models (Existing Methods)
E-PID and F-PID are typical systems for AI-CC parameter online-tuning with the intelligence of expert and fuzzy rules. The structure of the E-PID and F-PID control systems are shown in Figure  2. Compared with classical PID, the features and differences of E-PID and F-PID are as follows: According to the control parameter online-tuning rule, the control parameters , , of the PID controller will be updated in each adjustment cycle.
Definition of F-PID: If the control parameter online-tuning rule is the fuzzy calculation algorithm, the AI-CC system is called a fuzzy-reasoning-based method for parameter online tuning, which can be abbreviated as F-PID. Compared with classical PID, the features and differences of E-PID and F-PID are as follows: According to the control parameter online-tuning rule, the control parameters K p , K i , K d of the PID controller will be updated in each adjustment cycle.
Definition of F-PID: If the control parameter online-tuning rule is the fuzzy calculation algorithm, the AI-CC system is called a fuzzy-reasoning-based method for parameter online tuning, which can be abbreviated as F-PID. The online-tuned control parameters of F-PID are calculated as Equations (5) are outputs of fuzzy computing.
Definition of E-PID: If the control parameter online-tuning rule is the expert system algorithm, the AI-CC system is called an expert-rule-based parameter online-tuning control method, which can be abbreviated as E-PID.
The principle of the E-PID model is similar to that of the F-PID model: the control parameters of E-PID are calculated as Equations (8)-(10), -which means the parameters of E-PID can be obtained directly from the E-PID rule table.
The pseudocode of E-PID and F-PID control methods can be presented as Algorithm 2: Algorithm 2 E-PID (Expert system based PID) and F-PID (Fuzzy calculation based PID) control methods
End while 13. End

Application of E-PID and F-PID (Existing Application)
E-PID and F-PID are widely used in many applications because of its simple structure. E-PID was applied to a servo system by F. Kang, and the experimental results showed that E-PID could obtain an excellent control effect [23]. F-PID was applied to robot arm motion control by Ji, Y.L. [24]. F-PID has also been used to realize depth control for ROV (Remote Operated Vehicle) in a nuclear power plant.
As to an existing practical application of the E-PID and F-PID system, a heating control system of an oil production process is shown in Figure 3. (1) The three small green devices in the middle-right of Figure 3. are PID controllers. The small white devices on the right side are display and setting devices, which are connected to the sensors in the heating object and the PID controller. The computer is connected to the PID controller by an RS-485 data cable, which is designed to read the temperature of the controlled heating equipment and adjust the control parameters K p , K i , K d in each adjustment cycle. The cylindrical device on the right is the heating device; the input of this heating device is connected to the output of the PID controller to get the power voltage, which is monitored by the oscilloscope in the middle-left Figure 3. connected to the output of the PID controller to get the power voltage, which is monitored by the oscilloscope in the middle-left Figure 3.

EA-PID and FA-PID Models (Improved Methods)
The problems of E-PID and F-PID are as follows: The online-tuned parameters of E-PID and F-PID are set as fixed absolute values according to the expert rule table in E-PID and the fuzzy calculation formula in F-PID. Although , , can be updated in each adjustment cycle, the intelligence and performance features of E-PID and F-PID methods are limited.
In order to solve the above problems, EA-PID and FA-PID methods are proposed in this study. In EA-PID and FA-PID methods, the parameters are tuned by the change ratio instead of the absolute values in each adjustment cycle. Definition 3. EA-PID: In this study, EA-PID is an improved intelligent PID control method based on E-PID. Compared with E-PID, the ERT of E-PID is replaced by Change Ratio Table (

EA-PID and FA-PID Models (Improved Methods)
The problems of E-PID and F-PID are as follows: The online-tuned parameters of E-PID and F-PID are set as fixed absolute values according to the expert rule table in E-PID and the fuzzy calculation formula in F-PID. Although K p , K i , K d can be updated in each adjustment cycle, the intelligence and performance features of E-PID and F-PID methods are limited.
In order to solve the above problems, EA-PID and FA-PID methods are proposed in this study. In EA-PID and FA-PID methods, the parameters are tuned by the change ratio instead of the absolute values in each adjustment cycle.  The structure of the EA-PID and FA-PID control systems are shown in Figure 4.

Design of Fuzzy Controller
The fuzzy controller of F-PID and FA-PID is designed and shown in Figure 5. F-PID is designed as Figure 5a, and FA-PID is designed as Figure 5b.
(a) F-PID fuzzy controller parameter setting. Definition 5. Change ratio: The change ratio R(t) is a proportion between the control parameters at time t and t − 1 . The new control parameter is calculated as K(t) = K(t − 1)·R(t).

FA-PID Model
Kp (e(k), ec(k)) is the change ratio for K p at time k, which is the output of fuzzy calculation. The online-tuned control parameters of FA-PID at time t are calculated as Equations (11)-(13):

Design of Fuzzy Controller
The fuzzy controller of F-PID and FA-PID is designed and shown in Figure 5. F-PID is designed as Figure 5a, and FA-PID is designed as Figure 5b.

Improve F-PID to FA-PID
In FA-PID, ( ( ), ( )) is the change ratio for at time , which is the output of fuzzy calculation. The online-tuned control parameters of FA-PID at time are calculated as Equations (11)-(13):

Design of Fuzzy Controller
The fuzzy controller of F-PID and FA-PID is designed and shown in Figure 5. F-PID is designed as Figure 5a, and FA-PID is designed as Figure 5b.   A centroid is adopted as the method of defuzzification, which returns the center of gravity of the fuzzy set along the x-axis. The formula of the centroid is expressed as Equation (14):

EA-PID Model
3.2.1. Improve E-PID to EA-PID: Change ratios for updating , , of EA-PID, given by expert experience, can be designed, as in Table 1. For example, is one case of a change value when ( ) is very large and ( ) is very large. and are the boundary values.   A centroid is adopted as the method of defuzzification, which returns the center of gravity of the fuzzy set along the x-axis. The formula of the centroid is expressed as Equation (14):

Improve E-PID to EA-PID:
Change ratios for updating K p , K i , K d of EA-PID, given by expert experience, can be designed, as in Table 1. For example, R hh is one case of a change value when e(t) is very large and ec(t) is very large. E hign and E low are the boundary values.
, ec(t)) is the change ratio at time t. The control parameters of EA-PID are calculated as Equations (15)- (17): Table   The design of the rule table of the EA-PID algorithm is shown in Tables 2-4, which is similar to the rule table of the E-PID algorithm.

Implementation of EA-PID and FA-PID
In E-PID and F-PID methods, the values of the updated K p (t), K i (t), K d (t) are queried directly according to Equations (5)-(7) and Equations (8)-(10), respectively.
In EA-PID and FA-PID methods, the change ratios R p (t), R i (t), R d (t) are queried first. Then, the value of the updated K p (t), . Details are shown in Equations (11)- (13) and Equations (15)-(17) for each method, respectively.
The control system and AI-CC algorithms of EA-PID or FA-PID can be designed as Algorithm 3: Algorithm 3 EA-PID and FA-PID (which are defined in Definition 3 to 4) control methods
Input Expert rule table or Fuzzy rules; 4.
End while 13. End

Comparison of E-PID, F-PID, EA-PID, and FA-PID
All the comparative results of F-PID, E-PID, FA-PID, and EA-PID methods are listed in Table 5.
Change ratio for PID parameters: PID parameters: Change ratio for PID parameters: Tuning results

WNN-PID Model
The structure of the neural-network-based AI-CC system is shown in Figure 6. The difference between Figure 2 (E-PID and F-PID), Figure 4 (EA-PID and FA-PID), and Figure 6 is, in Figure 6, the parameters of the classic controller is tuned online by a neural network (online tuner).
Change ratio for PID parameters: PID parameters: Change ratio for PID parameters:

WNN-PID Model
The structure of the neural-network-based AI-CC system is shown in Figure 6. The difference between Figure 2 (E-PID and F-PID), Figure 4 (EA-PID and FA-PID), and Figure 6 is, in Figure 6, the parameters of the classic controller is tuned online by a neural network (online tuner). Begin 2.
Initialize NN weights W mn (t) and offsets b l (t); 4.
End while 13. End According to existing BPNN-PID and RBFNN-PID, the WNN-PID method is proposed as an online tuner in this study. Moreover, BPNN-PID and RBFNN-PID are selected as comparative algorithms to verify the advantages of the WNN-PID method.
The key to the NN-based parameter online-tuning algorithm is that finding the minimum mean of square error of system outputs is the training target of the neural network tuner. The target function can be expressed as Equation (18).
An alternative target function can be expressed as Equation (19). According to the simulation results, the different target functions will affect the results in numerical differences, but this does not affect the results and order of comparative simulations. Hence, both of the target functions can be adopted; all the NNs should apply the same target functions. In this study, the mean square error of Equation (18) is adopted as the target function for comparison.
Control parameters K p (t), K i (t), K d (t) are the result of the forward propagation of NNs according to Equations (20)- (27).
Ψ ab (t) is the scale transformation function of the hidden layers of WNN. The selection of Ψ ab (t) must satisfy the framework condition. O k (t) is the output of the output layer.

Online Tuning Algorithm of WNN-PID
Common training algorithms are the gradient descent method, Newton's method, the quasi-Newton method, and the conjugate gradient method. Among the above, the gradient descent method is the earliest, simplest, and most commonly used method. Hence, the gradient descent method is adopted as the training algorithm for BPNN-PID, RBFNN-PID, and WNN-PID methods.
The gradient descent method in the training algorithm can be designed as follows: w ij , ∆a j (t) and ∆b j (t) are changed to Equations (28)-(31).
Details of the above differentiation formulas are given in Equations (32)-(37):

1.
Anti-interference Configuration 1 for the saturation test.
(1) The max simulation time (max number of AC) is set as max_k = 8000 in order to ensure that the system can be stabilized, and the interference can maximize the effect. (2) The start time of interference is at d_k = 2000 (after the output reaches steady-state). (3) The interference duration is set as I _k = 6000 in order to test the features of saturation and convergence of the system output. (4) Interference intensity (amplitude or strength of disturbance) is set as dst_ u(k) = 0.5, which is added to the output of the PID controller. In summary: from simulation time 2000 to 6000, the control amount (output of PID controller) is changed to u(k) + dst_ u(k).
Simulation 1 is a saturation test that is based on basic configuration and Anti-interference Configuration 1. The results of the above 4 methods (E-PID, F-PID, EA-PID, and FA-PID) are shown in Figure 7.

Anti-interference Configuration 1 for the saturation test.
(1) The max simulation time (max number of AC) is set as max_ = 8000 in order to ensure that the system can be stabilized, and the interference can maximize the effect.

Anti-interference Configuration 2 for the unsaturation experiment.
(1) The max simulation time is set as max_k = 3000.
(2) The start time of interference is at d_k = 2000. (3) The interference duration is set as I _k = 10. (4) Interference intensity is set as dst_ u(k) = 0.5, which is added to the output of the PID controller. In summary: from simulation time 2000 to 2010, the control amount (output of PID controller) is changed to u(k) + dst_ u(k).
Simulation 2 is an unsaturation experiment that is based on basic configuration and Anti-interference Configuration 2. The results of the above 4 methods (E-PID, F-PID, EA-PID, and FA-PID) are shown in Figure 8. Simulation 2 is an unsaturation experiment that is based on basic configuration and Antiinterference Configuration 2. The results of the above 4 methods (E-PID, F-PID, EA-PID, and FA-PID) are shown in Figure 8. The above simulation results show that 1. The control speed of EA-PID is the best: The rising curve (especially when AC is at 0 to 500) in Figures 7 and 8 shows that E-PID and EA-PID are faster than F-PID and FA-PID. EA-PID has the fastest speed to reach a steady state (settling time of EA-PID is the smallest value). The speed of FA-PID is faster than F-PID (shorter rise time and steady time). Only EA-PID has overshot. 2. The steady-state error of EA-PID is the smallest. The steady-state error of FA-PID is smaller than F-PID, when AC is at 0 to 1500 in Figures 7 and 8. 3. EA-PID has the best ability of anti-interference, FA-PID has the second-best ability of antiinterference, E-PID is saturated under the influence of continuous interference, and F-PID is the worst performer (unstable, does not converge) when AC is at 2000 to 10,000 in Figure 7. 4. The anti-interference ability and recovery speed of EA-PID are the best. The recovery curve (when AC is at 2010 to 2300) in Figure 8 shows that the recovery time of EA-PID is the shortest.
The maximum values of system output in Simulation 1 (Figure 7) prove that the control system is unsaturation in Simulation 2 ( Figure 8) because the amplitude of system output in Simulation 1 (with longer time of interference) is larger than in Simulation 2.
In conclusion, most control performances of EA-PID are the best. Most control performances of FA-PID are better than F-PID. More simulation details of Figure 8 are listed in Table 6.
In Table 6, data in columns of rise time and settling time show the features of control speed. Data in the overshoot column show the overadjustment ability. Data in the steady-state error The above simulation results show that 1.
The control speed of EA-PID is the best: The rising curve (especially when AC is at 0 to 500) in Figures 7 and 8 shows that E-PID and EA-PID are faster than F-PID and FA-PID. EA-PID has the fastest speed to reach a steady state (settling time of EA-PID is the smallest value). The speed of FA-PID is faster than F-PID (shorter rise time and steady time). Only EA-PID has overshot. 2.
The steady-state error of EA-PID is the smallest. The steady-state error of FA-PID is smaller than F-PID, when AC is at 0 to 1500 in Figures 7 and 8.

3.
EA-PID has the best ability of anti-interference, FA-PID has the second-best ability of anti-interference, E-PID is saturated under the influence of continuous interference, and F-PID is the worst performer (unstable, does not converge) when AC is at 2000 to 10,000 in Figure 7.

4.
The anti-interference ability and recovery speed of EA-PID are the best. The recovery curve (when AC is at 2010 to 2300) in Figure 8 shows that the recovery time of EA-PID is the shortest.
The maximum values of system output in Simulation 1 (Figure 7) prove that the control system is unsaturation in Simulation 2 ( Figure 8) because the amplitude of system output in Simulation 1 (with longer time of interference) is larger than in Simulation 2.
In conclusion, most control performances of EA-PID are the best. Most control performances of FA-PID are better than F-PID. More simulation details of Figure 8 are listed in Table 6. In Table 6, data in columns of rise time T r and settling time T s show the features of control speed. Data in the overshoot column show the overadjustment ability. Data in the steady-state error column show the tracking ability of the control system. The best values are in bold and underlined in Table 6.  [26,27]. In each simulation group, the simulations are repeated 30 times. (6) The null hypothesis is that the difference in simulation results, such as rising time, settling time, and steady-state error, is more than 10%. The alternative hypothesis is that the difference in simulation results is less than 10%.
Secondly, the same configurations of all the above simulations are as follows: (1) In each AC (see Definition 1), parameters should be tuned once.
(2) The incremental digital PID algorithm is adopted as the controller model. (3) Second-order with delay is used as the controlled object model, which can be expressed as sys(s) = 200 s 2 +30s+1 ·e −τs . (4) Sampling period (AC interval) is set as ts = 1, which means that one program loop corresponds to one sample period. (5) Max simulation time is set as max_k = 3000, which is designed to be large enough to ensure that the system can be stabilized. (6) System input is set as rin(k) ≡ 1 for step response. (7) In order to make the experimental results more stable and convincing, each SC will be repeated 10 times.
The simulation results of the 10 repeated SC of BPNN-PID, RBFNN-PID, and WNN-PID are drawn in Figure 9. Curve of system input, output, error, output of controller, and parameters of K p , K i , K d simulation time are plotted.    In Figure 9, each SC is different from the others because the results of the initialization of the neural networks are different. As a result of different initialized weights and bias, the output of , , , , , are different in different SCs. The above results show that (1) all the NN-PID methods are verified as feasible: the output curves can approach system input and keep stable; (2) stability of WNN-PID and BPNN-PID are better than RBFNN-PID: unstable cases only happened in RBFNN-PID simulations. The unstable problem will be solved by stability guarantee methods in the following section of this study. In Figure 9, each SC is different from the others because the results of the initialization of the neural networks are different. As a result of different initialized weights and bias, the output of K p , K i , K d , u, yout, error are different in different SCs.
The above results show that (1) all the NN-PID methods are verified as feasible: the output curves can approach system input and keep stable; (2) stability of WNN-PID and BPNN-PID are better than RBFNN-PID: unstable cases only happened in RBFNN-PID simulations. The unstable problem will be solved by stability guarantee methods in the following section of this study.
The results of repeated 30 SCs, (each SC has 3000 ACs, max_k = 3000), of BPNN-PID, RBFNN-PID, and WNN-PID are shown in Table 7. The results of repeated 30 SCs, (each SC has 3000 ACs, _ = 3000), of BPNN-PID, RBFNN-PID, and WNN-PID are shown in Table 7.  Table 7 shows that (1) Table 8, which shows that the above conclusions have the same statistical characteristics. The difference in simulation results, such as rising time, settling time, and steady-state error, is smaller than 10% Table 8. Average results of the above three groups of simulations.

Analysis of Dynamic and Steady-State
The performance features of the above seven AI-CC methods (F-PID, FA-PID, E-PID, EA-PID BPNNPID, RBFNN-PID, and WNN-PID) are summarized. Data in columns of rising time and settling time show the features of control speed. Data in columns of steady-state error show the capability with output-tracking input of the system. In Table 9, EA-PID is focused on a comparison with all X-PID because all the features of EA-PID are the best. FA-PID is focused on a comparison with F-PID because FA-PID is improved in comparison to F-PID. WNN-PID is focused on a comparison with BPNN-PID because WNN-PID is improved in comparison to BPNN-PID. RBFNN-PID is not compared to BPNN-PID and WNN-PID because RBFNN-PID is not stable and a performance discussion of an unstable method is worthless.
Comparative results in Table 9 show that 1. Most features of EA-PID are the best because all the EA-PID result data (line 2) in Table 9 have the smallest values. Therefore, the improvement of EA-PID is successful. 2. FA-PID is better than F-PID because all the FA-PID result data (line 4) in Table 9 are smaller values than F-PID (line 3). Therefore, an improvement from F-PID to FA-PID is successful. The results of repeated 30 SCs, (each SC has 3000 ACs, _ = 3000), of BPNN-PID, RBFNN-PID, and WNN-PID are shown in Table 7.  Table 7 shows that (1) the control speed of WNN-PID is faster than BPNN-PID according to rise time ( ) and time of settling time ( ). Some of the control speeds of RBFNN-PID are faster than WNN-PID and BPNN-PID, but some of the control speeds of RBFNN-PID are unstable.
(2) The steady-state error of BPNN-PID is the smallest among all NN-PID methods.
Three additional groups of statistical tests are implemented. In each group of statistical tests, simulations are repeated 30 times. The average results of the above three groups of simulations are shown in Table 8, which shows that the above conclusions have the same statistical characteristics. The difference in simulation results, such as rising time, settling time, and steady-state error, is smaller than 10% Table 8. Average results of the above three groups of simulations.  Table 9, EA-PID is focused on a comparison with all X-PID because all the features of EA-PID are the best. FA-PID is focused on a comparison with F-PID because FA-PID is improved in comparison to F-PID. WNN-PID is focused on a comparison with BPNN-PID because WNN-PID is improved in comparison to BPNN-PID. RBFNN-PID is not compared to BPNN-PID and WNN-PID because RBFNN-PID is not stable and a performance discussion of an unstable method is worthless.

RBFNN
Comparative results in Table 9 show that 1. Most features of EA-PID are the best because all the EA-PID result data (line 2) in Table 9 have the smallest values. Therefore, the improvement of EA-PID is successful. 2. FA-PID is better than F-PID because all the FA-PID result data (line 4) in Table 9 are smaller values than F-PID (line 3). Therefore, an improvement from F-PID to FA-PID is successful. The results of repeated 30 SCs, (each SC has 3000 ACs, _ = 3000), of BPNN-PID, RBFNN-PID, and WNN-PID are shown in Table 7.  Table 7 shows that (1) the control speed of WNN-PID is faster than BPNN-PID according to rise time ( ) and time of settling time ( ). Some of the control speeds of RBFNN-PID are faster than WNN-PID and BPNN-PID, but some of the control speeds of RBFNN-PID are unstable.
(2) The steady-state error of BPNN-PID is the smallest among all NN-PID methods.
Three additional groups of statistical tests are implemented. In each group of statistical tests, simulations are repeated 30 times. The average results of the above three groups of simulations are shown in Table 8, which shows that the above conclusions have the same statistical characteristics. The difference in simulation results, such as rising time, settling time, and steady-state error, is smaller than 10% Table 8. Average results of the above three groups of simulations.  Table 9, EA-PID is focused on a comparison with all X-PID because all the features of EA-PID are the best. FA-PID is focused on a comparison with F-PID because FA-PID is improved in comparison to F-PID. WNN-PID is focused on a comparison with BPNN-PID because WNN-PID is improved in comparison to BPNN-PID. RBFNN-PID is not compared to BPNN-PID and WNN-PID because RBFNN-PID is not stable and a performance discussion of an unstable method is worthless.

RBFNN
Comparative results in Table 9 show that 1. Most features of EA-PID are the best because all the EA-PID result data (line 2) in Table 9 have the smallest values. Therefore, the improvement of EA-PID is successful. 2. FA-PID is better than F-PID because all the FA-PID result data (line 4) in Table 9 are smaller values than F-PID (line 3). Therefore, an improvement from F-PID to FA-PID is successful. The results of repeated 30 SCs, (each SC has 3000 ACs, _ = 3000), of BPNN-PID, RBFNN-PID, and WNN-PID are shown in Table 7.  Table 7 shows that (1) the control speed of WNN-PID is faster than BPNN-PID according to rise time ( ) and time of settling time ( ). Some of the control speeds of RBFNN-PID are faster than WNN-PID and BPNN-PID, but some of the control speeds of RBFNN-PID are unstable.
(2) The steady-state error of BPNN-PID is the smallest among all NN-PID methods.
Three additional groups of statistical tests are implemented. In each group of statistical tests, simulations are repeated 30 times. The average results of the above three groups of simulations are shown in Table 8, which shows that the above conclusions have the same statistical characteristics. The difference in simulation results, such as rising time, settling time, and steady-state error, is smaller than 10% Table 8. Average results of the above three groups of simulations.  Table 9, EA-PID is focused on a comparison with all X-PID because all the features of EA-PID are the best. FA-PID is focused on a comparison with F-PID because FA-PID is improved in comparison to F-PID. WNN-PID is focused on a comparison with BPNN-PID because WNN-PID is improved in comparison to BPNN-PID. RBFNN-PID is not compared to BPNN-PID and WNN-PID because RBFNN-PID is not stable and a performance discussion of an unstable method is worthless.

RBFNN
Comparative results in Table 9 show that 1. Most features of EA-PID are the best because all the EA-PID result data (line 2) in Table 9 have the smallest values. Therefore, the improvement of EA-PID is successful. 2. FA-PID is better than F-PID because all the FA-PID result data (line 4) in Table 9 are smaller values than F-PID (line 3). Therefore, an improvement from F-PID to FA-PID is successful.  Table 7 shows that (1) the control speed of WNN-PID is faster than BPNN-PID according to rise time (T r ) and time of settling time (T s ). Some of the control speeds of RBFNN-PID are faster than WNN-PID and BPNN-PID, but some of the control speeds of RBFNN-PID are unstable.
(2) The steady-state error of BPNN-PID is the smallest among all NN-PID methods.
Three additional groups of statistical tests are implemented. In each group of statistical tests, simulations are repeated 30 times. The average results of the above three groups of simulations are shown in Table 8, which shows that the above conclusions have the same statistical characteristics. The difference in simulation results, such as rising time, settling time, and steady-state error, is smaller than 10% Table 8. Average results of the above three groups of simulations.

BPNN -PID
T r The performance features of the above seven AI-CC methods (F-PID, FA-PID, E-PID, EA-PID BPNNPID, RBFNN-PID, and WNN-PID) are summarized. Data in columns of rising time and settling time show the features of control speed. Data in columns of steady-state error show the capability with output-tracking input of the system. In Table 9, EA-PID is focused on a comparison with all X-PID because all the features of EA-PID are the best. FA-PID is focused on a comparison with F-PID because FA-PID is improved in comparison to F-PID. WNN-PID is focused on a comparison with BPNN-PID because WNN-PID is improved in comparison to BPNN-PID. RBFNN-PID is not compared to BPNN-PID and WNN-PID because RBFNN-PID is not stable and a performance discussion of an unstable method is worthless.

WNN -PID St-Err
The simulation results plotted in Figure 10 show that most AI-CC systems have an acceptable anti-interference ability except RBFNN-PID. Some curves in the RBFNN-PID plots are not convergent (unable to reach steady-state after receiving interference). However, this problem is solved in the RBFNN-S-PID system by adding the stability guarantee algorithm, which is proposed in the previous section. The comparative details such as control speed etc. will be listed and discussed in the conclusion chapter.

Basis of Theoretical Analysis
From the above discussion, there are two types of AI-CC: (1) E-PID, EA-PID, F-PID, and FA-PID, composed of expert or fuzzy intelligence methods and the classical PID control method; (2) BPNN-PID, RBFNN-PID, and WNN-PID, which are the other type of AI-CC, composed of neural network intelligent methods and the classical PID control method.
All the above two types of AI-CC methods are based on the classical PID method, so all the classical theories for the PID control method can be used for system analysis, such as stability and nonlinear features.
The scope of research on controlled objects is limited to the scope of application of the classical PID control method. The linear time-invariant and approximate linear time-invariant systems will be discussed in this study because the PID control methods are widely and maturely used for linear time-invariant practical application systems. Stability analysis methods for high-order systems, delay systems, and inertial systems are discussed. A specific stability analysis case of a second-order delay system is provided.
According to the classical PID theories, theoretical analysis and simulation of AI-CC can also be applied to multiorder systems with delay linear time-invariant and approximate linear time-invariant

Basis of Theoretical Analysis
From the above discussion, there are two types of AI-CC: (1) E-PID, EA-PID, F-PID, and FA-PID, composed of expert or fuzzy intelligence methods and the classical PID control method; (2) BPNN-PID, RBFNN-PID, and WNN-PID, which are the other type of AI-CC, composed of neural network intelligent methods and the classical PID control method.
All the above two types of AI-CC methods are based on the classical PID method, so all the classical theories for the PID control method can be used for system analysis, such as stability and nonlinear features.
The scope of research on controlled objects is limited to the scope of application of the classical PID control method. The linear time-invariant and approximate linear time-invariant systems will be discussed in this study because the PID control methods are widely and maturely used for linear Processes 2020, 8, x FOR PEER REVIEW 24 of 30 Figure 11. 10 SCs simulation for unstable RBFNN-PID systems.

AI-CC-S method is proposed
The AI-CC-S method is based on the equivalent criterion of the Nyquist stability criterion. All the control parameters that are adjusted by the AI-CC method should be judged according to the AI-CC-S method.
For example, if the controlled object model can be drawn as In each AC, amplitude and phase are calculated, the pseudocode is provided as follows, which can be implemented by different programming languages: The bode plots of the control system with the latest ( ), ( ), ( ) can be plotted, as in Figure 12. According to the above theoretical analysis, the current online-tuned parameters ( ), ( ), ( ) can make the system work steadily, so the current online-tuned parameters ( ), ( ), ( ) can be adopted (not abandoned) for the PID controller.

AI-CC-S method is proposed
The AI-CC-S method is based on the equivalent criterion of the Nyquist stability criterion. All the control parameters that are adjusted by the AI-CC method should be judged according to the AI-CC-S method.
For example, if the controlled object model can be drawn as G Obj (s) = 200 s 2 +30s+1 ·e −s , the controller model is G Ctr (s) = k p + k i s + k d ·s = k d ·s 2 +k p ·s+k i s , and the current time is t. The current parameters at time t are as follows: K p (t) = 2, K i (t) = 0.2, and K d (t) = 0.2, which are pretested to make the system run steadily.
As the open-loop transfer function is G Open (s), the amplitude margin can be calculated as Gm = 20·log 10 (gm) = −0.156dB < 0, and the phase margin can be calculated as Pm = pm = 9.36 deg > −π.
In each AC, amplitude and phase are calculated, the pseudocode is provided as follows, which can be implemented by different programming languages:

AI-CC-S method is proposed
The AI-CC-S method is based on the equivalent criterion of the Nyquist stability criterion. All the control parameters that are adjusted by the AI-CC method should be judged according to the AI-CC-S method.
For example, if the controlled object model can be drawn as As the open-loop transfer function is , the amplitude margin can be calculated as 20 • 0.156dB 0, and the phase margin can be calculated as 9.36deg .
In each AC, amplitude and phase are calculated, the pseudocode is provided as follows, which can be implemented by different programming languages: The bode plots of the control system with the latest , , can be plotted, as in Figure 12. According to the above theoretical analysis, the current online-tuned parameters , , can make the system work steadily, so the current online-tuned parameters , , can be adopted (not abandoned) for the PID controller.
The bode plots of the control system with the latest K p (t), K i (t), K d (t) can be plotted, as in Figure 12. According to the above theoretical analysis, the current online-tuned parameters K p (t), K i (t), K d (t). can make the system work steadily, so the current online-tuned parameters K p (t), K i (t), K d (t) can be adopted (not abandoned) for the PID controller.

RBFNN-S-PID is proposed
RBFNN-S-PID is a method of RBFNN-PID composed with the AI-CC-S method. The effect of the improved RBFNN-S-PID method is shown in Figure 13.
Throughout the online-tuning process, all updated parameters are judged. RBFNN-PID is unstable, but RBFNN-S-PID with the AI-CC-S method is stable. From the above theoretical analysis and simulation, the stability problem is solved.

3.
RBFNN-S-PID is proposed RBFNN-S-PID is a method of RBFNN-PID composed with the AI-CC-S method. The effect of the improved RBFNN-S-PID method is shown in Figure 13.

RBFNN-S-PID is proposed
RBFNN-S-PID is a method of RBFNN-PID composed with the AI-CC-S method. The effect of the improved RBFNN-S-PID method is shown in Figure 13.
Throughout the online-tuning process, all updated parameters are judged. RBFNN-PID is unstable, but RBFNN-S-PID with the AI-CC-S method is stable. From the above theoretical analysis and simulation, the stability problem is solved.  Comparing Figure 11 with Figure 13, the RBFNN-S-PID system is much more stable: (1) All the RBFNN-S-PID results are convergent (finally stabilized); (2) RBFNN-S-PID has fewer disadvantages.
Throughout the online-tuning process, all updated parameters are judged. RBFNN-PID is unstable, but RBFNN-S-PID with the AI-CC-S method is stable. From the above theoretical analysis and simulation, the stability problem is solved.

Conclusions
In this study, three AI-CC methods (EA-PID, FA-PID, and WNN-PID) are proposed, and theoretical analysis for the stability of the above methods and stability guarantee methods are provided. Then, problems of manual-based methods are solved, and the performance of existing AI-CC methods is improved.