Fuzzy Controllers With Neural Network Predictor for Second-Order Linear Systems With Time Delay

Artificial intelligence methods are widely applied in advanced control practices such as fuzzy calculation-based intelligent control method, neural network-based predictive control method, etc. Control performances are expected to be improved such as: faster control speed, smaller steady-state error, and less repeated manual tuning workloads in harmful environments for engineers. Main works are as follows: Firstly, change ratio-based fuzzy adjusted PID (FA-PID) method is improved. The adjusted parameters of FA-PID are the multiplication result of the change ratioa at the current control cycle and the control parameters at the time of previous adjustment cycle. Secondly, prediction of back propagation neural networks-based fuzzy-PID (BPNN-F-PID) and prediction of back propagation neural networks-based FA-PID (BPNN-FA-PID) are improved, in which the adjusted control parameters are calculated according to the predicted output of control system. Thirdly, comparative simulations of all the above methods are implemented, a series of better effects are found. Effects are as follows: The improved controllers have better performance such as faster control speed, better ability of anti-interference, restrained overshoot and smaller steady-state error.

SECTION I. Introduction

In practical control applications, classical control (CC) methods such as proportional integral derivative (PID) methods are widely used [1]. More than 90% control applications and processes of production are based on PID control [2], [3]. The PID method was proposed in the 1910s. In the following years, PID-based autotuning and online tuning methods have been developed and improved [4], [5]. Online tuning work for control parameters is necessary and important because the control systems cannot keep working with good performance [6], [7] without the online tuning process. Artificial intelligence (AI) has become state-of-the-art technology, which is widely applied in automatic control engineering. Therefore, AI-based improved control methods are widely concerned in practical engineering including the above 90% applications.

According to the above background, the motivations of this study are as follows: (1) Try to solve the control problems in the non-linear and time-varying environment which widely exist in the practical applications such as classical PID control, fuzzy control without PID, neural network based PID control [5], etc. (2) Try to improve the performance of existing classical PID and advanced fuzzy PID methods by increasing the control speed, reducing the steady-state error which are needed in most practical applications. (3) Try to increase the ability of anti-interference and suppress the overshoot of classical PID and Fuzzy PID, etc., because overshoot is not expected to appear in various applications. All the above motivations are similar as the previous published research such as [5].

There are some existing advanced control methods. In previous researches with different methods, some important conclusions [5] have been discussed in detail, which can be summarized as follows:

1. The predictive control method (PDC) has been widely used in industry since 1970s, and it has a good feature of strong robustness. Until now, most predictive control technologies are applied in linear control system, which rely on the experiences and ability of engineers to achieve a good control [8], [9]. It could help people to build dynamic closed-loop system with good stability in a limited time [10]. However, the predictive control for non-linear systems and stochastic uncertain systems is insufficient. The predictive control often requires more accurate models and higher computing performance [11].

2. The adaptive control method (ADC) has been proved to design stable, safe, and automated systems since it was proposed in the last century [12]. In current applications, the characteristics of the disturbance are usually unknown and time-varying [13]. The adaptive control method can continuously identify model parameters and adjust control parameters to satisfy the changes of the work environment [14].

   However, it is theoretically difficult to obtain a general solution, and the process of parameter estimation takes a long time to converge. Measurement errors in practical applications will directly affect the parameters of the controller. Direct adaptive control adjusts control gain based on error calculation. Therefore, it is difficult to design a nonlinear adaptive control system.

3. The intelligent control method (AIC) includes fuzzy logic control [15], [16], neural network control, expert control, hierarchical intelligent control [17], etc. The advantages of AIC are obvious in control parameters online tuning and adjusting which can be used to solve the problems of non-linear and complex control. In addition, AIC has good ability of fault tolerance and functions to correct some errors [18], [19]. However, a simple intelligence-based control is often unable to control well because the control systems are complex, unknown, and dynamic. The theoretical analysis of intelligent control methods such as fuzzy control is very difficult because the models of them are complex and difficult to analyze with existing methods.

4. Artificial intelligence (AI)-based improved classical control (CC) methods (AI-CC) use AI methods to adjust control parameters of CC online.

AI-CC is valuable to be studied [20] because of the following reasons: (1) AI-CC methods are based on the classical control theory so most classical control theories can be used for theoretical analysis such as dynamic, non-linear and complex features analysis. (2) AI-CC methods have both advantages of intelligent methods and classical control methods [20]. (3) AI-CC methods have more practical values because they are based on the classical control methods which already have a wide range of applications.

A. Related Work

The existing AI-CC methods are as follows: (1) Expert system-based self-tuning PID (E-PID) [22], (2) Fuzzy reasoning-based self-tuning PID (F-PID) [23], (3) Genetic Algorithms-based self-tuning PID (GA-PID) [24], (4) Neural network (NN)-based PID (NN-PID) methods such as: Back propagation neural networks-based self-tuning PID (BPNN-PID) [25], Radial basis function neural networks-based self-tuning PID (RBFNN-PID) [26], Wavelet neural networks-based self-tuning PID (WNN-PID) [27], etc. In the previous research [5], the above NN-PID-based methods and improvements are discussed. In this study, the above (2) F-PID method is focused, the reason is as follows:

The problems of the above existing AI-CC methods are as follows: (1) In E-PID, instability occurs because of sudden changes of the control parameters and disturbances. Undesirable overshoot occurs in some practical applications [28]; (2) In NN-PID, the structure of the neural network is complex, and the speed of training is slow [29]; (3) The dynamic features, steady-state error and other performances of both F-PID and NN-PID need to be further improved.

Therefore, this study is focused on F-PID because of the following reasons: Firstly, compared with the expert rules-based reasoning method of E-PID, fuzzy calculation can express more complex control logic. Secondly, compared with NN-PID, the fuzzy calculation is faster, while the neural networks are complex, and the training process of neural networks costs more time.

The comparisons among the related fuzzy control methods are as follows: (1) A suitable controller was designed based on the extension of Kessler’s methods in the inner loop, supported by predictive technique and PID-fuzzy controller in the outer loop [31], [32]. This method has good performances in parallel distributed control. (2) A stable state-feedback fuzzy controller was designed based on a neuro-fuzzy state-space model. This model is successfully trained using the experimental data acquired from a real robotic arm [33]. (3) A new approach to the stable design of Takagi–Sugeno–Kang fuzzy logic control systems was supported by a novel stability analysis theorem. This theorem is based on previous results concerning the application of Lyapunov’s direct method for a general class of chaotic systems [34]. (4) A hybrid controller was designed based on Magneto-rheological damper lookup table for quarter car suspension [35]. Its effectiveness has been proved in controlling vehicle suspension system. (5) A PID-type fuzzy logic controller tuning strategy based on direct fuzzy relations is proposed to compute the PID constants. It helps a motion control algorithm for robots to reach a desired position decreasing the steady error and energy consumption which shows that performances such as saturation time and overshoot of F-PID are better than classical PID [1]. (6) A fuzzy PID controller with nonlinear compensation term is designed, in which the parameters of membership functions are optimized by PSO algorithm. It shows that F-PID can achieve satisfactory control effect in areas like suppressing random disturbances comparing with classical PID [36]. (7) A Takagi-Sugeno fuzzy PID control is proposed for variable pitch system and the PI parameters of the current loop are set by using the internal model principle. It shows that simple fuzzy reasoning system combined with PID control will form a compound control method which have better stability and dynamic performance [37].

Comparing with adaptive fuzzy control (AFC) methods, the problems of the non-adaptive fuzzy control (N-AFC) methods are as follows: N-AFC methods need to configure parameters before the control process, and those parameters are usually obtained by the experience of engineers. However, the precise parameters given by engineers might not be the optimal parameters, and they cannot be adjusted during the control process to adapt to the changes of control environment so the control effect cannot keep best. Therefore, this study is focused on AFC methods: AFC methods do not need accurate configurations of control parameters before the control process. AFC can automatically adjust parameters during the process to obtain the better values which will make the control process keep better performances.

The specific class of control system issecond-order linear systems with time delay which is also described in detail at the beginning of Section II in order to show that our idea is general under our assumptions, which have been discussed in the previous work in reference [5]. Structures, descriptions, equations and pseudocodes of all existing, improved methods, simulations, results, and analysis are provided in the following sections.

B. Contributions

Main contribution of this study are as follows:

1. FA-PID is the improvement of F-PID method. In F-PID method, control parameters are calculated directly by fuzzy calculation. Change ratios and control parameters of last control cycle are multiplied to adjust PID parameters, which makes FA-PID has better performance than F-PID.

2. FA-PID is improved based on F-PID, BPNN-F-PID is improved based on F-PID, and BPNN-FA-PID is improved based on FA-PID. In order to reduce the overshoot of the control system in some special situations, BPNN predictor is considered. Both BPNN-F-PID and BPNN-FA-PID use the predicted output of control system to adjust control parameters, which can improve the anti-interference ability of F-PID and FA-PID.

3. Comparative simulations of all the existing and improved methods are implemented in order to find the improvements. Two types of simulation are designed: unsaturated experiment which has short-term interference and saturated experiment which has long-term interference.According to the results of the above two types of simulation, better performance of the control system such as speed, overshoot and anti-interference are found.

The differences between the improved methods and existing methods are as follows: (1) The adjusted control parameters of FA-PID are calculated according to change ratios and the historical values of control parameters, while the adjusted control parameters of F-PID are queried directly from the fuzzy rules. (2) The adjusted control parameters of BPNN-FA-PID are calculated according to the predicted output of control system, while the adjusted control parameters of FA-PID are calculated according to the current system output.

The rest of the paper is organized as follows: In Section II, existing PID and F-PID models are described. In Section III, the improved FA-PID models are described. In Section IV, the improved BPNN-F-PID model and the improved BPNN-FA-PID model are described. In Section V, the comparative simulations of all the existing and improved methods are implemented in order to find advantages and disadvantages. In Section VI, conclusions and future work are discussed.

SECTION II. F-PID Model (Existing Model)

In the previous section, the background, motivation, contribution, and structure of this article are introduced. In this section, existing models such as PID and F-PID are described which are the basis of the improved methods of this study. This section is organized as follows: In part A, the existing PID model is described in detail. In part B, the existing F-PID model is described in detail.

Assumptions 1:

The controlled plant of the model has the characteristics of second-order delay.

Assumptions 2:

The PID parameters of the controller can be adjusted in each operation cycle.

Assumptions 3:

The disturbance in the system can be calculated to the input of the controlled plant so that the disturbance can be added to the next round of tuning.

Definition 1 Adjustment cycle (AC):

‘1 AC’ is only one control process at one fixed simulation time (e.g. $t\mathrm {=3}$ ). In each AC, control parameters $K_{p}(t)$ , $K_{i}(t)$ , $K_{d}(t)$ , the control amount $u(t)$ , system output $y_{out}(t)$ , and system error $error(t)$ , $d_{error}(t)$ are adjusted once.

Definition 2 Change ratio:

Change ratios $R_{Kp}^{FA}(t)$ , $R_{Ki}^{FA}(t)$ and $\mathrm { }R_{Kd}^{FA}(t)$ are proportions between the control parameters at time $t$ and $t-1$ . The adjusted control parameter is calculated as: $K_{p}\left ({t }\right)=K_{p}(t-1)\cdot R_{Kp}^{FA}(t)$ .

A. Classical PID Model (Existing Method)

PID control is a classical control method, which realizes control process according to the error of system output and applies the correction based on proportional, integral, and derivative terms. The classical PID model has been studied [5], and the important conclusion is summarized as follows:

The structure of classical PID system is shown in Fig. 1. At time $t$ , system error and the differential result of error (i.e. e and de/dt, respectively) are calculated according to system input and output. The PID controller will calculate the amount of control $u$ , which is the input of controlled plant. In the classical PID method, the control parameters $K_{p}$ , $K_{i}$ , $K_{d}$ are constants, which are set to fixed values before the system starts. Finally, the system output is calculated.

FIGURE 1.

System structure of classical PID control systems.

The model of classical PID controller is defined as (1): $$\begin{equation*} u(t)=K_{p}\cdot \left({e\left ({t }\right)+\frac {1}{K_{i}}\int {e\left ({t }\right)} dt+K_{d}\frac {de\left ({t }\right)}{dt}}\right)\tag{1}\end{equation*}$$

At time $t$ , the updated $u\left ({t }\right)$ is calculated according to incremental PID method $u\left ({t }\right)=u\left ({t-1 }\right)\mathrm {+\Delta }u(t)$ , which is expressed as (2): $$\begin{align*}&\hspace {-0.5pc}\Delta u(t)=K_{p}(t)\left ({e\left ({t }\right)-e\left ({t-1 }\right) }\right)+K_{i}(t)e\left ({t }\right) \\&\qquad \qquad \qquad \displaystyle {+K_{d}(t)\left ({e\left ({t }\right)-2e\left ({t-1 }\right)+e\left ({t-2 }\right) }\right)} \tag{2}\end{align*}$$

The output of control system $y_{out}\left ({t }\right)$ , which is the output of the controlled plant, can be calculated according to the digital PID method of discrete systems. Firstly, the transfer function of the controlled plant is discretized to $a\left ({i }\right)$ and $b\left ({j }\right)$ . Secondly, $y_{out}\left ({t }\right)$ is calculated as (3): $$\begin{align*} y_{out}\left ({t }\right)=&-a\left ({2 }\right)y\left ({t-1 }\right)-a\left ({3 }\right)y\left ({t-2 }\right)-a\left ({4 }\right)y\left ({t-3 }\right) \\&+\,b\left ({2 }\right)u\left ({t-1 }\right)\!+\!b\left ({3 }\right)u\left ({t-2 }\right)\!+\!b\left ({4 }\right)u\left ({t-3 }\right) \\\tag{3}\end{align*}$$

Pseudocode of PID control method is provided as Algorithm 1:

Algorithm 1 The Existing Classical PID Method

1.

Begin

2.

Initialize PID control system;

3.

While ($t\le max\_{}t$ ) do:

4.

Simulation time updated: t = t + 1;

5.

System input $r_{in}\left ({t }\right)$ is updated;

6.

Calculate $\Delta u\left ({t }\right)$ and $u\left ({t }\right)$ according to (1) and (2);

7.

Calculate $y_{out}(t)$ according to (3);

8.

Calculate $e\left ({t }\right)=y_{out}\left ({t }\right)-r_{in}(t)$ ;

9.

End while

10.

End

B. F-PID Model (Existing Method)

1) Structure of F-PID

Fuzzy PID (F-PID) is a typical AI-CC method. The structure of the F-PID control system is shown in Fig. 2: (1) The input of control system is $r_{in}(t)$ , the control target at time $t$ . (2) The output of control system is $y_{out}(t)$ , the control result (output of controlled plant) at time $t$ . (3) The online tuner of F-PID is based on Fuzzy Reasoning Calculation (FRC). FRC is shown inside the dotted line area in Fig. 2. The input of FRC is system error $e(t)$ and the differential of system error $\mathrm {ec}(t)$ . The adjusted $K_{p}(t)$ , $K_{i}(t)$ and $K_{d}(t)$ are the output of FRC.

FIGURE 2.

System structure of F-PID system.

The control variable $u(t)$ is calculated as (4) in each AC, which is the output of PID controller and the input of controlled plant: $$\begin{equation*} u(t)=K_{p}(t)\cdot (e\left ({t }\right)+\frac {1}{K_{i}\left ({t }\right)}\int {e\left ({t }\right)} dt+K_{d}(t)\frac {de\left ({t }\right)}{dt}\tag{4}\end{equation*}$$

2) Design of FRC

FRC has three parts which are shown in the dotted line area in Fig. 2: (1) Input of Fuzzy Calculation, (2) Fuzzy Calculation and (3) Output of Fuzzy Calculation.

The general design of FRC are shown in Fig. 3, which are specifically described in the following 3 parts.

FIGURE 3.

Design of Fuzzy controller.

In the first row of Fig. 3, the first sub-figures show the relationship of the above 3 parts. The following 5 sub-figures in the first and second rows of Fig. 3 show the member functions of fuzzified and defuzzified calculations. In the last row of Fig. 3, the relationship between the inputs and outputs are shown.

1. Input of Fuzzy Calculation:

   The inputs of fuzzy controller are $e_{f}\left ({t }\right)$ , the fuzzified result of system error $e(t)$ , and ${ec}_{f}\left ({t }\right)$ , the fuzzified result of differential of system error $ec(t)$ . $e(t)$ and $ec(t)$ are also the inputs of FRC.

   The above fuzzified results are calculated based on the membership functions which are provided as (5) and (6): $$\begin{align*} e_{f}\left ({t }\right)=&\mathrm {Fuzzified\_{}e}(e(t)) \tag{5}\\ {ec}_{f}\left ({t }\right)=&\mathrm {Fuzzified\_{}ec}(ec(t))\tag{6}\end{align*}$$

   The membership functions are listed in TABLE 1.$\mathrm {Fuzzified\_{}e()}$ can be expressed as $e$ {NB(Negative Big), Z(Zero), PB(Positive Big)}. The membership function $\mathrm {Fuzzified\_{}ec()}$ can be expressed as $ec$ {NB(Negative Big), Z(Zero), PB(Positive Big)}.

   In TABLE 1, “zmf”, “trimf” and “smf” stand for Z-membership function, triangular membership function, and S-membership function, respectively. The data in the TABLE 1 is the knee point of each variable. For example, as for the input variable 1, the knee points for three membership functions are [−3,1], [−2,0,2], and [−1,3].

2. Fuzzy Calculation:

   The design principles of fuzzy rules of fuzzy calculation are based on the principles of PID control which can be listed as follows: Firstly, when $e \boldsymbol {(t)}$ is a large value, $K_{p} \boldsymbol {(t)}$ should be set to a large value in order to get shorter rising time of control process, and $K_{d} \boldsymbol {(t)}$ should be set to small value in order to reduce the overshoot. Secondly, when $e \boldsymbol {(t)}$ is a large value, smaller $K_{p} \boldsymbol {(t)}$ and larger $K_{i} \boldsymbol {(t)}$ should be adopted. Thirdly, when $ec \boldsymbol {(t)}$ is small, $K_{p} \boldsymbol {(t)}$ and $K_{i} \boldsymbol {(t)}$ should be set to large values for better stability.

   According to the above principles, the fuzzy calculation of F-PID is designed based on the fuzzy rules which can be listed in “If-Then” format as follows:

   1. If ($e_{f}$ is NB) and (${ec}_{f}$ is NB) then ($K_{pf}$ is PB)($K_{if}$ is NB)($K_{df}$ is Z)

   2. If ($e_{f}$ is NB) and (${ec}_{f}$ is Z) then ($K_{pf}$ is PB)($K_{if}$ is NB)($K_{df}$ is NB)

   3. If ($e_{f}$ is NB) and (${ec}_{f}$ is PB) then ($K_{pf}$ is Z)($K_{if}$ is Z)($K_{df}$ is Z)

   4. If ($e_{f}$ is Z) and (${ec}_{f}$ is NB) then ($K_{pf}$ is PB)($K_{if}$ is NB)($K_{df}$ is Z)

   5. If ($e_{f}$ is Z) and (${ec}_{f}$ is Z) then ($K_{pf}$ is Z)($K_{if}$ is Z)($K_{df}$ is Z)

   6. If ($e_{f}$ is Z) and (${ec}_{f}$ is PB) then ($K_{pf}$ is NB)($K_{if}$ is PB)($K_{df}$ is Z)

   7. If ($e_{f}$ is PB) and (${ec}_{f}$ is NB) then ($K_{pf}$ is Z)($K_{if}$ is Z)($K_{df}$ is PB)

   8. If ($e_{f}$ is PB) and (${ec}_{f}$ is Z) then ($K_{pf}$ is NB)($K_{if}$ is PB)($K_{df}$ is PB)

   9. If ($e_{f}$ is PB) and (${ec}_{f}$ is PB) then ($K_{pf}$ is NB)($K_{if}$ is PB)($K_{df}$ is PB)

3. Output of Fuzzy Calculation:

   The outputs of fuzzy calculation are fuzzified results of $K_{pf}(t)$ , $K_{if}(t)$ and $K_{df}(t)$ . The adjusted results of $K_{p}(t)$ , $K_{i}(t)$ and $K_{d}(t)$ are the defuzzified results of $K_{pf}(t)$ , $K_{if}(t)$ and $K_{df}\left ({t }\right)$ respectively, according to (7) to (9): $$\begin{align*} K_{p}(t)=&\mathrm {Defuzzified\_{}}K_{p}(K_{pf}(t)) \tag{7}\\ K_{i}(t)=&\mathrm {Defuzzified\_{}}K_{i}(K_{if}(t)) \tag{8}\\ K_{d}(t)=&\mathrm {Defuzzified\_{}}K_{d}(K_{df}(t))\tag{9}\end{align*}$$

TABLE 1 The Values of Fuzzy Rules for F-PID

The membership functions of $\mathrm {Defuzzified\_{}}K_{p}()$ , $\mathrm {Defuzzified\_{}}K_{i}()$ , and $\mathrm {Defuzzified\_{}}K_{d}()$ are listed in TABLE 1.$K_{p}(t)$ , $K_{i}(t)$ , and $K_{d}\left ({t }\right)$ are the output of FRC.

The membership function $\mathrm {Defuzzified\_{}}K_{p}\mathrm {()}$ can be expressed as $K_{p}$ {NB(Negative Big), Z(Zero), PB(Positive Big)}. The membership function $\mathrm {Defuzzified\_{}}K_{i}\mathrm {()}$ can be expressed as $K_{i}$ {NB(Negative Big), Z(Zero), PB(Positive Big)}. The membership function $\mathrm {Defuzzified\_{}}K_{d}\mathrm {()}$ can be expressed as $K_{d}$ {NB(Negative Big), Z(Zero), PB(Positive Big)}.

$K_{p}(t)$ , $K_{i}(t)$ and $K_{d}(t)$ are defuzzied by the calculation of defuzzification. Centroid is one type of defuzzification method, which returns the x-axis coordinate of gravity in the fuzzy set. The formula of centroid defuzzification method is expressed as (10)–(12), where $\mu (x_{i})$ is the membership function, and the output of $\mu (x_{i})$ is the fuzzified result of input $x_{i}$ . $$\begin{align*} K_{p}(t)=&\frac {\sum \nolimits _{j=1}^{n} {\mu _{Kp}\left ({x_{j} }\right)\mathrm {.}x_{j}} }{\sum \nolimits _{j=1}^{n} {\mu _{Kp}\left ({x_{j} }\right)}} \tag{10}\\ K_{i}(t)=&\frac {\sum \nolimits _{j=1}^{n} {\mu _{Ki}\left ({x_{j} }\right)\mathrm {.}x_{j}} }{\sum \nolimits _{j=1}^{n} {\mu _{Ki}\left ({x_{j} }\right)}} \tag{11}\\ K_{d}(t)=&\frac {\sum \nolimits _{j=1}^{n} {\mu _{Kd}\left ({x_{j} }\right)\mathrm {.}x_{j}} }{\sum \nolimits _{j=1}^{n} {\mu _{Kd}\left ({x_{j} }\right)}}\tag{12}\end{align*}$$

3) Implement of F-PID

The pseudocode of F-PID can be presented as Algorithm 2:

Algorithm 2 The Existing F-PID Method

1.

Begin

2.

Initialize X-PID control system model;

3.

Input Fuzzy Rules’ Table;

4.

While ($t\le max\_{}t$ ) do:

5.

Simulation time updated: t = t + 1;

6.

System input $r_{in}\left ({t }\right)$ is updated;

7.

Calculate $e(t)$ and $ec(t)$ ;

8.

Calculate $K_{p}(t\mathrm {),}K_{i}(t\mathrm {),}K_{d}(t)$ by FRC and TABLE 1;

9.

Calculate $\Delta u\left ({t }\right)$ and $u\left ({t }\right)$ according to (2) and (4);

10.

Calculate $y_{out}(t)$ according to (3);

11.

End while

12.

End

SECTION III. FA-PID Model (Improved Methods)

In the previous section, the F-PID method is introduced. In this section, FA-PID is the improvement of F-PID to make the performance of the existing control method better. This section is organized as follows: In part A, the structure of FA-PID model is described in detail. In part B, the change ratio of FA-PID is explained. In part C, the fuzzy controller is designed. Part D is the implement of FA-PID. The researched contents of Section III to V are the core parts of improved methods which are different from all the previous series research of our AI-CC research group.

A. Structure of FA-PID

F-PID is improved by change ratio-based fuzzy adjusted PID control (FA-PID) method. In FA-PID method, control parameters at time $t$ are adjusted according to the result of multiplying change ratio and parameters at time $t-1$ , while the control parameters of F-PID at time $t$ are adjusted according to the result of fuzzy calculation.

The system structure of FA-PID methods is shown in Fig. 4.

FIGURE 4.

System structure of FA-PID methods.

B. Calculation of Change Ratio

Main differences between FA-PID and F-PID are specifically described as follows: (1) In F-PID method, the adjusted $K_{p}(t)$ , $K_{i}(t)$ and $K_{d}(t)$ are FRC. (2) In FA-PID method, the change ratios $R_{p}(t)$ , $R_{i}(t)$ and $R_{d}(t)$ are calculated firstly.

Then, values of the updated $K_{p}(t)$ , $K_{i}(t)$ and $K_{d}(t)$ are calculated as $K_{p}(t-1)\cdot R_{p}(t)$ , $K_{i}(t-1)\cdot R_{i}(t)$ and $K_{d}(t-1)\cdot R_{d}(t)$ , respectively, according to (13) to (15).

In FA-PID, $R_{p}(t)$ is the change ratio of $K_{p}$ at time $t$ , which is the output of FRC. The online tuned control parameters of FA-PID are calculated as (13) to (15): $$\begin{align*} K_{p}(t)=&R_{p}(t\mathrm {)\cdot }K_{p}(t\mathrm {-1)} \tag{13}\\ K_{i}(t)=&R_{i}(t\mathrm {)\cdot }K_{i}(t\mathrm {-1)} \tag{14}\\ K_{d}(t)=&R_{d}(t\mathrm {)\cdot }K_{d}(t\mathrm {-1)}\tag{15}\end{align*}$$

C. Design of Fuzzy Calculation

FRC of FA-PID is also designed based on fuzzy rules, which is similar to the FRC of F-PID.

1. Input of Fuzzy Calculation:

   The input of fuzzy controller of FA-PID are also $e_{f}\left ({t }\right)$ , the fuzzified result of system error $ec(t)$ , and ${ec}_{f}\left ({t }\right)$ , the fuzzified result of differential of system error $ec(t)$ . $e(t)$ and $ec(t)$ are also the inputs of FRC of FA-PID. The fuzzified results are calculated based on the membership functions as (5) and (6)

   The values of membership functions FA-PID are provided and shown in TABLE 2 and Fig. 5.

2. Fuzzy Calculation:

   The fuzzy calculation of FA-PID is designed based on the fuzzy rules which can be listed in “If-Then” format, which is similar as F-PID as follows:

   1. If ($e_{f}$ is NB) and (${ec}_{f}$ is NB) then ($R_{pf}$ is PB)($R_{if}$ is NB)($R_{df}$ is Z)

   2. If ($e_{f}$ is NB) and (${ec}_{f}$ is Z) then ($R_{pf}$ is PB)($R_{if}$ is NB)($R_{df}$ is NB)

   3. If ($e_{f}$ is NB) and (${ec}_{f}$ is PB) then ($R_{pf}$ is Z)($R_{if}$ is Z)($R_{df}$ is Z)

   4. If ($e_{f}$ is Z) and (${ec}_{f}$ is NB) then ($R_{pf}$ is PB)($R_{if}$ is NB)($R_{df}$ is Z)

   5. If ($e_{f}$ is Z) and (${ec}_{f}$ is Z) then ($R_{pf}$ is Z)($R_{if}$ is Z)($R_{df}$ is Z)

   6. If ($e_{f}$ is Z) and (${ec}_{f}$ is PB) then ($R_{pf}$ is NB)($R$ is PB)($R_{df}$ is Z)

   7. If ($e_{f}$ is PB) and (${ec}_{f}$ is NB) then ($R_{pf}$ is Z)($R$ is Z)($R_{df}$ is PB)

   8. If ($e_{f}$ is PB) and (${ec}_{f}$ is Z) then ($R_{pf}$ is NB)($R$ is PB)($R_{df}$ is PB)

   9. If ($e_{f}$ is PB) and (${ec}_{f}$ is PB) then ($R_{pf}$ is NB)($R$ is PB)($R_{df}$ is PB)

3. Output of Fuzzy Calculation:

   The output of fuzzy calculation of FA-PID are fuzzified results of $R_{pf}(t)$ , $R_{if}(t)$ , and $R_{df}(t)$ . The defuzzification method of FA-PID is similar to the defuzzification method of F-PID, which is based on centroid method. The defuzzified calculation are according to (16)–(18): $$\begin{align*} R_{p}(t)=&\frac {\sum \nolimits _{j=1}^{n} {\mu _{Rp}\left ({x_{j} }\right)\mathrm {.}x_{j}} }{\sum \nolimits _{j=1}^{n} {\mu _{Rp}\left ({x_{j} }\right)}} \tag{16}\\ R_{i}(t)=&\frac {\sum \nolimits _{j=1}^{n} {\mu _{Ri}\left ({x_{j} }\right)\mathrm {.}x_{j}} }{\sum \nolimits _{j=1}^{n} {\mu _{Ri}\left ({x_{j} }\right)}} \tag{17}\\ R_{d}(t)=&\frac {\sum \nolimits _{j=1}^{n} {\mu _{Rd}\left ({x_{j} }\right)\mathrm {.}x_{j}} }{\sum \nolimits _{j=1}^{n} {\mu _{Rd}\left ({x_{j} }\right)}}\tag{18}\end{align*}$$

TABLE 2 The Value of Fuzzy Rules for FA-PID

FIGURE 5.

Design of Fuzzy controller.

The adjusted results of $K_{p}(t)$ , $K_{i}(t)$ , and $K_{d}(t)$ are the defuzzified results of $K_{pf}(t)$ , $K_{if}(t)$ , and $K_{df}\left ({t }\right)$ respectively, according to (7) to (9).

D. Implement of FA-PID

The AI-CC algorithms of FA-PID can be designed as Algorithm 3:

Algorithm 3 FA-PID

1.

Begin

2.

Initialize XA-PID control system model;

3.

Input Fuzzy Rules’ Table;

4.

While ($t\le max\_{}t$ ) do:

5.

Simulation time updated: $t=t +1$ ;

6.

System input $r_{in}\left ({t }\right)$ is updated;

7.

Calculate $e(t)$ and $ec(t)$ ;

8.

$R_{p}\left ({t }\right)$ , $R_{i}\left ({t }\right)$ , $R_{d}\left ({t }\right)$ are calculated;

9.

$K_{p}\left ({t }\right)$ , $K_{i}\left ({t }\right)$ , $K_{d}\left ({t }\right)$ are adjusted by (13) to (15);

10.

Calculate $\Delta u\left ({t }\right)$ and $u\left ({t }\right)$ according to (1) and (2);

11.

Calculate $y_{out}\left ({t }\right)$ according to (3);

12.

End while

13.

End

SECTION IV. Design of BPNN-F-PID and BPNN-FA-PID

In the previous section, the FA-PID method are introduced. In this section the BPNN prediction-based F-PID (BPNN-F-PID) method and BPNN prediction-based FA-PID (BPNN-FA-PID) are introduced. This section is organized as follows: In part A, the structure of BPNN-F-PID model and BPNN-FA-PID model are described in detail. In part B, the input and output of BPNN Predictor are explained. In part C, the calculation of BPNN Predictor is introduced. In part D, BPNN-F-PID and BPNN-FA-PID are implemented. In part E, the differences of the above four methods are compared.

Assumption in this study: It is assumed that the performance of control system will be better (the control parameters are better adjusted) in two conditions: (1) The current output $y_{out}(t)$ at time $t$ is replaced by the predicted future output $y_{out}(t+p)$ at time $t+p$ ; (2) The output of the control system is predicted accurately; (3) The predicted output of the control system has more valuable information for parameter adjustment work according to the existed predictive control methods.

Training, validating, and testing processes of BPNN are as follows:

1. Training: The features of the training set are the history values of status of control system, which can be expressed as: [$r_{in}(t)$ , $u(t\mathrm {), }y_{out}(t)$ , $k_{p}(t)$ , $k_{i}(t)$ , $k_{d}(t)$ ], [$r_{in}(t\mathrm {-1)}$ , $u(t\mathrm {-1), }y_{out}(t\mathrm {-1)}$ , $k_{p}(t\mathrm {-1)}$ , $k_{i}(t\mathrm {-1)}$ , $k_{d}(t\mathrm {-1)}$ ],…,[$r_{in}(t-N)$ , $u(t-N\mathrm {), }y_{out}(t-N)$ , $k_{p}(t-N)$ , $k_{i}(t-N)$ , $k_{d}(t-N)$ ]. The label of trainset is $y_{out}(t+M)$ . So, after training, the predictor can predict $y_{predict}(t+M)$ . The trainset is from various control processes.

2. Validating: According to the above successful training, the predicted results can be validated by more control processes. Comparing each predicted result of $y_{predict}(t+M)$ and the real values of label $y_{out}(t+M)$ , the errors can be found and judged.

3. Testing: In the applications of control process, after the 100th control cycles, there are enough [$r_{in}(t)$ , $u(t\mathrm {), }y_{out}(t)$ , $k_{p}(t)$ , $k_{i}(t)$ , $k_{d}(t)$ ], [$r_{in}(t\mathrm {-1)}$ , $u(t\mathrm {-1), }y_{out}(t\mathrm {-1)}$ , $k_{p}(t\mathrm {-1)}$ , $k_{i}(t\mathrm {-1)}$ , $k_{d}(t\mathrm {-1)}$ ],…, [$r_{in}(t-N)$ , $u(t-N\mathrm {), }y_{out}(t-N)$ , $k_{p}(t-N)$ , $k_{i}(t-N)$ , $k_{d}(t-N)$ ] to train the predictor to predict $y_{out}(t+M)$ .

A. Structure of BPNN-F-PID and BPNN-FA-PID

Structure of BPNN-F-PID control system can be drawn as Fig. 6. The difference between Fig. 2 and Fig. 6 is that: control parameters of classical controller will be tuned online by BP neural network. BP neural network is the online tuner.

FIGURE 6.

Structure diagram of BPNN-F-PID control system.

Structure of BPNN-FA-PID control system can be drawn as Fig. 7. The difference between Fig. 4 and Fig. 7 is the same as the difference between F-PID and FA-PID: FRC for F-PID is replaced by FRC for FA-PID.

FIGURE 7.

Structure diagram of BPNN-FA-PID control system.

B. Input and Output of BPNN Predictor

The inputs of BPNN predictor (time series with historical information) and outputs of control system are as follows: $r_{in}$ , $K_{p}$ , $K_{i}$ , $K_{d}$ , $u$ , $y_{out}$ , which are not only at time $t$ but also at $t-3$ and $t-6$ . The above time series can be expressed as a vector time series as (19): $$\begin{align*}&X(t)=\{r_{in}\left ({t }\right),K_{p}\left ({t }\right),K_{i}\left ({t }\right),K_{d}\left ({t }\right),u\left ({t }\right),y_{out}\left ({t }\right), \\&{r_{in}\left ({t-3 }\right),K}_{p}\left ({t-3 }\right),K_{i}\left ({t-3 }\right),K_{d}\left ({t-3 }\right), \\&u\left ({t-3 }\right),y_{out}\left ({t-3 }\right),r_{in}\left ({t-6 }\right)\mathrm {, }K_{p}\left ({t-6 }\right), \\&K_{i}\left ({t-6 }\right),K_{d}\left ({t-6 }\right),u\left ({t-6 }\right)\mathrm {, }y_{out}\left ({t-6 }\right)\}\tag{19}\end{align*}$$

In (19), $X\left ({t }\right)$ is the input of the BPNN prediction algorithm, which contain information of $r_{in}$ , $K_{p}$ , $K_{i}$ , $K_{d}$ , $u$ , $y_{out}$ at time $t-6$ , $t-3$ , and $t$ . $X\left ({t }\right)$ is also the input of training samples of predictor. In BPNN predictor, the label of training samples at time $t$ is $Y\left ({t }\right)=y_{out}\left ({t+10 }\right)$ .

The output of BP predictor at future time $t+10$ can be expressed as (20). $$\begin{equation*} y_{predict}\left ({t }\right)=O_{k}=\sigma \left ({\sum {\sigma \left ({\sum {X{\left ({t }\right)W}_{ij}} }\right)W_{jk}} }\right)\tag{20}\end{equation*}$$

$X\left ({t }\right)$ is the input of simulation samples, including $r_{in}$ , $K_{p}$ , $K_{i}$ , $K_{d}$ , $u$ , $y_{out}$ at time $t,t-3$ ,and $t-6$ . $y_{predict}\left ({t }\right)$ is the predictive result at time $t+10$ . $W_{ij}$ and $W_{jk}$ are the weights between each layer of BPNN. $O_{k}$ is the output of BPNN. In (20), $k=1$ because there is only one output.

C. Calculation of BPNN Predictor

The target of BPNN-F-PID is to get the minimum error of the control system. The system error is affected by the control parameters, and the control parameters are tuned by the BPNN according to the weights of the BPNN. The target function can be expressed as (21). $$\begin{equation*} \min E(t)=\frac {1}{2}(r_{in}(t)-y_{out}(t))^{2}\tag{21}\end{equation*}$$

An alternative target function can be expressed as (22). According to the simulation results, the effects of different target functions are similar so both above target functions can be adopted. All the simulations should apply the same target functions. In this study, mean square error of (22) is adopted as the target function. $$\begin{equation*} \min E(t)=|r_{in}(t)-y_{out}(t)|\tag{22}\end{equation*}$$

Formulas of training process of BPNN are expressed as follows: $$\begin{align*} W_{ij}\left ({t+1 }\right)=&W_{ij}\left ({t }\right)\mathrm {+\Delta }W_{ij}\left ({t }\right)+\beta \Delta W_{ij}\left ({t-1 }\right)\tag{23}\\ W_{jk}\left ({t+1 }\right)=&W_{jk}\left ({t }\right)\mathrm {+\Delta }W_{jk}\left ({t }\right)+\beta \Delta W_{jk}\left ({t-1 }\right)\tag{24}\\ \Delta W_{jk}\left ({t }\right)=&-\eta \left ({O_{k}-t_{k} }\right)O_{k}\left ({1-O_{k} }\right)O_{j}\tag{25}\\ \Delta W_{ij}\left ({t }\right)=&-\eta O_{i}O_{j}\left ({1-O_{j} }\right) \\&\times \,\sum \limits _{k=1}^{K} {\left ({O_{k}-t_{k} }\right)O_{k}\left ({1-O_{k} }\right)W_{jk}}\tag{26}\\ O_{k}=&\sigma \left ({\sum \limits _{j=1}^{J} {\sigma \left ({\sum \limits _{i=1}^{I} {X(t\mathrm {)\cdot }W_{ij}} }\right)\cdot W_{jk}} }\right)\qquad \tag{27}\\ O_{i}=&X(t)\tag{28}\\ O_{j}=&\sigma \left ({O_{i}\cdot W_{ij} }\right)\tag{29}\end{align*}$$

In (29), $\sigma \left ({x }\right)=\frac {1}{1+e^{-x}}$ is the sigmoid activation function in the hidden and output layer of BPNN. $O_{i}$ , $O_{j}$ and $O_{k}$ are the outputs of each layer of BPNN. The explanation of each variable in BPNN are shown in TABLE 3.

TABLE 3 Meanings for Symbols of BPNN

D. Implement of BPNN-F-PID and BPNN-FA-PID

The algorithms of BPNN-F-PID and BPNN-FA-PID can be designed as Algorithm 4:

Algorithm 4 BPNN-F-PID and BPNN-FA-PID

1.

Begin

2.

Initialize the control system of BPNN-F-PID;

3.

Initialize BPNN weights $W_{ij}\left ({t }\right)$ and $W_{jk}\left ({t }\right)$ ;

4.

While $t\le max\_{}t$ do:

5.

Simulation time is updated: $t =t+1$ ;

6.

System input $r_{in}\left ({t }\right)$ is updated;

7.

If $predict\_{}begin==1$ do:

8.

If $predict\_{}firstTime ==1$ do:

9.

Create BPNN $model$ ;

10.

End If

11.

Train BPNN $model$ according to (23)–(29);

12.

Calculate $y_{predict}\left ({t }\right)$ according to (19)–​(20);

13.

Calculate $error(t\mathbf {)}$ according to $y_{predict}\left ({t }\right)$ ;

14.

Else

15.

Calculate $error(t\mathbf {)}$ according to $y_{out}(t)$ ;

16.

End If

17.

Training of BPNN: update $W_{ij}\left ({t }\right)$ and $W_{jk}\left ({t }\right)$ according to (23) to (29);

18a.

$K_{p}$ , $K_{i}$ , $K_{d}$ are calculated by FRC and TABLE 1;

18b.

$R_{p}$ , $R_{i}$ , $R_{d}$ are calculated by FRC and TABLE 2;

19a.

$K_{p}\left ({t }\right)$ , $K_{i}\left ({t }\right)$ , $K_{d}\left ({t }\right)$ of BPNN-F-PID are adjusted according to FRC of F-PID;

19b.

$K_{p}\left ({t }\right)$ , $K_{i}\left ({t }\right)$ , $K_{d}\left ({t }\right)$ of BPNN-FA-PID are adjusted according to (13)–(15);

20.

Calculate $\boldsymbol {\Delta }u\left ({t }\right)$ and $u\left ({t }\right)$ according to (1)–​(2);

21.

Calculate $y_{out}(t)$ according to (3);

22

End while

23.

End

E. Comparison Among F-PID, FA-PID, BPNN-F-PID and BPNN-FA-PID

The differences among F-PID, FA-PID, BPNN-F-PID, and BPNN-FA-PID are as follows:

1. Function of adjustment: F-PID adjusts parameters according to the rule table only. FA-PID uses the change ratio from the old parameters and the rule table. BPNN-F-PID and BPNN-FA-PID add the BPNN predictor based on the F-PID and FA-PID.

2. Basis of adjustment: Both of F-PID and BPNN-F-PID are based on the calculation of the fuzzy rules, while the FA-PID and BPNN-FA-PID are based on the change ratio and fuzzy rules.

3. Calculation of adjustment: The adjustment of F-PID and BPNN-F-PID are the output of the controller. The adjustment of FA-PID and BPNN-FA-PID are the result of the output times the old parameter.

SECTION V. Simulation

In the previous section, the existing method and improved method are introduced. In this section the simulations of those methods are introduced and the result of simulations are compared to show the improvement of the new method improved in this article. This section is organized as follows: In part A, two types of simulations are described. In part B, saturated simulations of F-PID, FA-PID, BPNN-F-PID, and BPNN-FA-PID methods are implemented to verify the effects of the improvements. In part C, unsaturated simulation is implemented.

A. Design of Simulation

Two types of simulations are designed in this study: (1) Unsaturated simulation, which has a short-term interference, (2) Saturated simulation, which has a long-term interference. The second simulation proves that the output of the first simulation is unsaturated and shows different results under the short-term and long-term interferences. Configurations of F-PID and FA-PID systems should be set similarly.

Basic configuration for both saturated and unsaturated simulations: (1) In each AC (see Definition 1), parameters should be tuned one time. (2) Incremental digital PID algorithm is adopted as the controller model. (3) Second-order with delay is used as the controlled plant model, which can be expressed as $sys\left ({s }\right)=\frac {250}{2s^{2}+30s+1}\mathrm {\cdot }e^{\mathrm {-1.5s}}$ . (4) Sampling period (AC interval) is set as $ts=1$ , which means that one program loop corresponds to one sample period. (5) Initial value of each control parameter is set as $K_{p}\left ({0 }\right)\mathrm {=1,}K_{i}\left ({0 }\right)\mathrm {=0.1}$ , and $K_{d}\left ({0 }\right)\mathrm {=0}$ , which are verified to keep the system running steadily. (6) System input is set as $r_{in}(k\mathrm {)\equiv 1}$ for step response.

B. Results Analysis of Saturated Simulation

Configuration for saturated simulation: (1) The max simulation time (max number of AC) is set as $\mathrm {max\_{}}k\mathrm {=10000}$ 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 $d\_{}k\mathrm {=3000}$ (after the output reaching steady state). (3) The interference duration is set as $I\mathrm { \_{}}k\mathrm {=4000}$ , in order to simulate the saturation and convergence of the system output. (4) Interference intensity (amplitude or strength of disturbance) is set as $dst\_{} u(k\mathrm {)=0.5}$ , which is added to the output of the PID controller. In summary: from simulation time 3000 to 7000, the control amount (output of PID controller) is changed to $u\left ({k }\right)+dst\_{} u(k)$ .

Comparative simulation results of F-PID, FA-PID, BPNN-F-PID, and BPNN-FA-PID control systems are shown in Fig. 8 (a) to (d) respectively. The change curves of system input, system output, system error, control variable and control parameters $K_{p}$ , $K_{i}$ , $K_{d}$ with simulation time are plotted.

FIGURE 8.

Simulation result of each algorithm.

The above results show that: (1) FA-PID and BP-FA-PID methods are verified to be feasible because the output curves can approach system input and keep stable. (2) F-PID and BP-F-PID cannot converge under the influence of interference.

According to the above simulations, the comparative results of simulation with the 4 methods (F-PID, FA-PID, BP-F-PID and BP-FA-PID) are shown in Fig. 9 (a) and (b)

FIGURE 9.

Results of saturated simulation of 4 AI-CC systems.

Fig. 9 shows the result of the saturated simulation. From the Fig. 9 (a) it can be found that the F-PID and the BP-F-PID do not converge after the long-term interference. That means F-PID and BP-F-PID are not stable after the long-term interference. In the unsaturated simulation, all the above four methods can converge under the short-term interference.

The result of the saturated simulation shows that: FA-PID and BPNN-FA-PID has the better performance of control speed (The curves of the FA-PID and the BPNN-FA-PID rise faster). Meanwhile, both of those two methods have better performance of anti-interference (converged after long-term interference).

C. Results Analysis of Unsaturated Simulation

Configuration for unsaturated simulation: (1) The max simulation time is set as $\mathrm {max\_{}}k\mathrm {=5000}$ . (2) The start time of interference is at $d\_{}k\mathrm {=3000}$ . (3) The interference duration is set as $I\mathrm { \_{}}k\mathrm {=50}$ . (4) Interference intensity is set as $dst\_{} u(k\mathrm {)=0.5}$ , which is added to the output of PID controller. In summary: from simulation time 3000 to 3050, the control amount (output of PID controller) is changed to $u\left ({k }\right)+dst\_{} u(k)$ .

Configuration for unsaturated simulation: The results of simulations with the 4 methods (F-PID, FA-PID, BP-F-PID and BP-FA-PID) are shown in Fig. 10 (a) and (b). The peaks (max values) of output curve of unsaturated simulation in Fig. 10 are lower than the peaks (top values) of output curve of saturated simulation in Fig. 9. This proves that the output of control system in unsaturated simulation in Fig. 10 are unsaturated.

FIGURE 10.

Results of unsaturated simulation of 4 AI-CC systems.

More simulation details of Fig. 10 are listed in TABLE 4. In TABLE 4, Rising Time $T_{r}$ and Settling Time $T_{s}$ show the features of control speed. Overshoot shows the over-adjustment ability. Steady-state error shows the tracking ability of the control system. Recovery time is used to represent recovery speed. The best values are bolded and underlined in TABLE 4.

TABLE 4 Comparison of Simulation Results of Control System Output and Error

According to the simulation results in Fig. 9, Fig. 10 and TABLE 4, findings are as follows:

1. FA-PID has improved the control speed of F-PID. The rising curves in Fig. 9(b) and Fig. 10(b) show that the control speed of FA-PID and BPNN-FA-PID are faster than the control speed of F-PID and BPNN-F-PID. FA-PID has the fastest speed to reach steady state (i.e. settling time of FA-PID is the smallest). The speed of BP-FA-PID (smaller rising time and settling time) is faster than BP-F-PID. More details are shown in TABLE 4: Control speed of FA-PID is the fastest (Tr = 269 and Ts = 324 are the smallest). BPNN-FA-PID (Tr = 275, Ts = 330) is faster than others. It means that FA-PID has faster control speed than F-PID. The reason of this improved effect is that: the range of adjusted control parameters of FA-PID are larger than F-PID. In another word, the possible larger values of control parameters of FA-PID are more than F-PID.

2. FA-PID has improved the steady-state error of F-PID. F-PID has the smallest steady-state error at the beginning because of the overshoot, but after around time equal to 1500 to 3000 in Fig. 10 (a), FA-PID has the smallest steady-state error. More details are shown in TABLE 4: The stability of FA-PID is the best (Steady-state error = 0.0037 is the smallest). BPNN-FA-PID (Steady-state error = 0.0064) is better than BPNN-F-PID (Steady-state error = 0.01115), which means BPNN-FA-PID has smaller Steady-state error than BPNN-F-PID. The reason of this improved effect is same as the above first conclusion.

3. The overshoot of the control system of BPNN prediction-based methods are smaller. Overshoot is restrained by BPNN. BPNN-FA-PID method has the smallest overshoot shown in Fig. 9(a) and Fig. 10(a), which means it can restrain the overshoot in some cases. More details are shown in TABLE 4: The overshoot of BPNN-FA-PID is the best (overshoot = 0.0074 is the smallest). BPNN-F-PID (overshoot = 0.0076) is better than FA-PID (overshoot = 0.0087). Therefore, the BPNN-FA-PID is the best one for some industry processes compared with other methods. Those industry processes require control variable to approach the plant value gradually. The overshoot of BPNN-F-PID is smaller than that of F-PID and FA-PID. The reason of this improved effect is that: the unexpected larger output values of control system (such as overshoot) can be predicted by BPNN based predictor so the control parameters are adjusted in advance according to the predicted output of control system.

4. BPNN-F-PID and BPNN-FA-PID have improved the ability of anti-interference of F-PID and FA-PID. BPNN-F-PID has the best recover speed in short term shown in Fig. 10(a). The recovery time of BPNN-F-PID in short term is the shortest. BPNN-FA-PID has the second good ability of anti-interference. There is no saturation under the influence of continuous interference. F-PID and BPNN-F-PID are the worst in long term interference (i.e. they are unstable and cannot converge) shown in Fig. 9(a). The reason of this improved effect is same as the above third conclusion.

5. Results (output of system) of unsaturated simulation is unsaturated so the results are valid: comparing the results between unsaturated simulation and saturated simulation which are shown in Fig. 9(a) and Fig. 10(a). It can be found that the system output of unsaturated simulation is unsaturated because the maximum output of unsaturated simulation (with longer time of interference) is larger than the maximum system output of saturated simulation.

In conclusion, according to unsaturated and saturated simulations, most performances of BPNN-FA-PID are better than BPNN-F-PID. Most performances of FA-PID are better than F-PID and BPNN-F-PID. The effects (faster control speed, smaller steady-state error, smaller overshoot, and stronger ability of anti-interference) of the improvements are better.

SECTION VI. Conclusion

In the previous section, related work, existing and improved methods, comparative simulations are discussed. In this section, all the improved effects are compared together, and the future work are discussed based on the comparison of related and latest research of references.

The comparative results of F-PID, FA-PID, BPNN-F-PID, and BPNN-FA-PID simulations are summarized in TABLE 5. Comparative indicators are designed as follows: (1) Control speed is measured by $T_{r}$ and $T_{s}$ . (2) Tracking ability is measured by Steady-state error. Therefore, the intelligent level of AI-CC system and the ability requirements of engineers can be evaluated.

TABLE 5 Comparison of Different Intelligent Control Systems

In this study, F-PID is improved by 3 methods (FA-PID, BPNN-F-PID, and BPNN-FA-PID). The adjustment rules of control parameters of F-PID are replaced by the change ratio rules of FA-PID. In BPNN-F-PID and BPNN-FA-PID, the predictor of BPNN neural network are adopted to predict the system output. $e$ and $ec$ can be calculated according to the result of predictor. After that, control parameters can be obtained in the rule table of F-PID and improved rule table of FA-PID respectively.

Comparative simulation results among F-PID, FA-PID, BPNN-F-PID, and BPNN-FA-PID methods can be summarized in TABLE 5.

According to TABLE 5, conclusions are summarized as follows: (1) The improvements of FA-PID are as follows: The control speed of FA-PID is faster than F-PID, BPNN-F-PID and BPNN-FA-PID. The steady-state error of FA-PID is smaller than that of F-PID and BPNN-F-PID. The overshoot of FA-PID is smaller than that of F-PID. It means that the performance of F-PID is improved by FA-PID. (2) The improvements of BPNN-FA-PID are as follows: BPNN-FA-PID can suppress the overshoot, and it is stable under long term interference.

Comparing other related fuzzy control methods with improved methods of this study, many advantages are adopted in this study to improve the PID and F-PID methods with faster control speed, smaller steady-state error, smaller overshoot, and better ability of anti-interference. Details of comparisons and adoptions are as follows: (1) Reference [31] and [32] provide the high quality control signals and sensory feedback to facilitate surgery over the time-delay network. The above idea is adopted to this study, and the structure and controlled plant are based on the classical PID control model, which make the old practical applications easy and convenient to upgrade. (2) Reference [33] improves the transient performance during initial stages and suitably switches to the original stable controller to utilize its better performance during steady-state conditions. The above idea is adopted to this study. Excellent (optimal) initialized configurations and fuzzy rules are not necessary to be configured before the system goes into production. The improved method can adjust and optimize control parameters in the production process by change ratio. The scale of adjusted control parameters of FA-PID are wider than F-PID. (3) Reference [34] shows a serious advantage with respect to LMI-based fuzzy logic control system design approaches that involve control laws, which are sensitive to all process parameters. The above idea is adopted to this study, and the improved methods can be also applied to wide practical applications because the classical PID method is widely used. (4) Reference [35] shows that the proposed Hybrid control lookup table provides better vibration isolation capability. The above idea is adopted in this study. Two types of fuzzy table are designed to calculate the control parameters of F-PID and the change ratio of FA-PID. Some improved effects can be found in the comparative results.

In the future: Existing industrial applications of PID systems will be upgraded to FA-PID and BPNN-FA-PID systems. Performances such as faster control speed, lower steady-state error, etc., will be better improved by adopting more intelligent AI methods such as deep learning, etc.