# Neural Network Supervision Control Strategy for Inverted Pendulum Tracking Control

Authors: Hongliang Gao; Xiaoling Li; Chao Gao; Jie Wu
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

This paper presents several control methods and realizes the stable tracking for the inverted pendulum system. Based on the advantages of RBF and traditional PID, a novel PID controller based on the RBF neural network supervision control method (PID-RBF) is proposed. This method realizes the adaptive adjustment of the stable tracking signal of the system. Furthermore, an improved PID controller based on RBF neural network supervision control strategy (IPID-RBF) is presented. This control strategy adopts the supervision control method of feed-forward and feedback. The response speed of the system is further improved, and the overshoot of the tracking signal is further reduced. The tracking control simulation of the inverted pendulum system under three different signals is given to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Inverted pendulum system has been widely investigated in the past few decades based on two important characteristics of high order and strong coupling, which are important problems in control field. And it is an unstable, nonlinear, and multivariable system. Inverted pendulum control methods have a wide range of applications in military, aerospace, robotics, and general industrial processes, such as balancing problems during robot walking, verticality issues during rocket launch, and attitude control issues during satellite flight. The RBF neural network learning control algorithm has been a hot topic in current academic research. This algorithm can solve nonlinear problems, tracking problems, and external interference problems. Therefore, using this method to study the tracking problem of the inverted pendulum is of great significance.The RBF (radial basis function) neural network controllers of the nonlinear system are designed based on proportion integration differentiation (PID), and these methods have good control results. Here are some related research results. In [1], it is presented that RBF network to estimate complex and precise dynamics mainly solves the problem of uncertainty and external interference in the context of complex space. This method is used to solve the problem of model uncertainty and input error. In [2], a neural network adaptive control algorithm with PID is proposed. The self-learning ability and self-adaptation uncertain system dynamic characteristics are used to significantly reduce the impact of resistance disturbance on speed. The system has strong robustness under the parameter variations and external disturbances. In [3], a scheme which combines a proportion differentiation control and a RBF neural network adaptive control algorithm is used. Among them, it uses the PD control to track the trajectory of the end effector of the wire-driven parallel robot (WDPR). The RBF neural network control algorithm is applied to approximate parameters. The combination of these two methods reduces the approximation error, enhances the robustness, and improves the accuracy of the WDPR. In [4], a fuzzy logic-based offline control strategy of a single-wheeled inverted pendulum robot (SWIPR) is presented to study the error, system set time, and rise time, etc. In the end, a good control effect was achieved. In [5], RBF network is applied to reduce chatter and increase stability. In [6], it proposes a single-layer nonlinear controller to achieve that the inverted pendulum can be adjusted to a stable state from any initial position and achieves four degrees of freedom. Hossein [7] put forward a kind of improved PID control method based on RBF neural network. It shows that the algorithm has a better control effect than the traditional PID for the tension. In [8, 9], adaptive control methods are applied to the input or output of nonlinear systems. They are, respectively, applicable to a class of nonsmooth nonlinear systems and a class of multi-input multioutput nonlinear time-delay systems with input saturation. In [10, 11], two different adaptive methods were proposed to solve the unknown and stochastic nonlinear tracking problems of nonlinear systems. And other scholars have studied the stabilization and convergence of the system [12–14]. Kumar et al. [15] use adaptive control technology to deal with tracking problems. At present, many researchers [16–20] have conducted in-depth studies on various inverted pendulum models using different control methods.In fact, no matter what the control system is, the system will tend to be stable under the action of the controller. Therefore, RBF neural network is used as a model for approximating uncertainty, to study error convergence, and achieved good convergence results in [21]. In [22], it is proposed to use the RBF algorithm to estimate the residual error, reduce the error through the design of the controller, increase the control signal, and obtain good heating requirements. In [23], the dynamic characteristics of machine system are explored and the characteristics of the RBF neural network are used to study the tracking problem. Under the large structure of the RBF neural network, different basis functions are selected for a comparative study to eliminate chattering in [24]. In [25], the RBF neural network implements self-feedback control, accurate prediction, and real-time control of reasonable data. It has improved tracking accuracy and estimated unmodeled dynamics and external interference issues in [26]. As more and more academic researchers understand the approximation characteristics of RBF, they add RBF neural networks to various fields to study the dynamic characteristics of different systems. It mainly solves the problems of nonlinearity, uncertainty, and external interference and uses the Lyapunov function to ensure the effectiveness of the algorithm, so that it reduces system errors and reaches a stable state [27–34].In recent years, the footprint of PID applications can be seen in different fields. From simple PID control algorithm to complex PID algorithm control, it has played a role in different control fields. In [35, 36], the PID control algorithm and operation rules are studied, respectively. The characteristics of the PID algorithm are explored, and the PID has a certain degree of adaptability through simulation. According to the characteristics of the PID algorithm, some scholars have studied the tuning of the PID controller [37, 38]. Tuning PID parameters are used to optimize system performance according to actual conditions. As PID parameter tuning technology becomes more and more mature, some interested scholars use PID as a controller to study system stability and tracking issues [39–42]. In order to adapt PID to more situations, some scholars have launched the control research of fuzzy PID [43, 44] and fractional PID [45, 46]. It can be seen that the PID control algorithm is a relatively classic control method.According to the references, we can find that the RBF neural network control algorithm basically uses the Lyapunov function to determine the stability conditions. In [47], based on the nonlinear U model, RBF neural network and PD parallel control algorithm are proposed. The Lyapunov function determines the conditions of system stability, and under this condition, the tracking effect has been improved. However, from the tracking effect, the error between the system output and the tracking signal is large, and the tracking situation with external interference is not considered. Therefore, in this article, we consider these problems based on the inverted pendulum model to study its tracking problem.In short, the control of the inverted pendulum model mainly includes three major control performances, namely, stability, accuracy, and rapidity. Then, for the tracking problem of the inverted pendulum, the three comprehensive performances also need to be considered. Therefore, we designed a supervision control method PID-RBF. We further design another supervision control strategy IPID-RBF. The stable tracking of the signal is achieved by supervision control strategy. In general, the main innovations of this paper include the following:(1)
PID-RBF strategy ensures the stability of the system. The overshoot of the system is reduced and the robustness of the system is enhanced. In the case of interference, the parameters can be adjusted adaptively to control signal tracking.(2)
IPID-RBF strategy further solves the problem of large overshoot in the control process. The adjustment time of the system is further reduced. This strategy has strong anti-interference ability, fast stability, and small error with the tracking curve.(3)
In the control process, we can use the PID-RBF strategy to replace the traditional PID control strategy. This way can make the system overshoot smaller and system stability better. IPID-RBF strategy further improves the overall performance of the system. In the IPID-RBF control strategy, the system has a faster response speed, better stability, and robustness.The rest of the paper is organized as follows. The relevant control objective is presented in Section2. Neural network supervision control design is presented in Section 3. The simulation study is discussed in Section 4. Finally, the conclusions are given in Section 5.


## 2. Control Objective

The inverted pendulum system can be difficult to control as the order increases. At the same time, the inverted pendulum system itself has the characteristics of complexity, instability, and nonlinearity. The inverted pendulum system is often used as an experimental project in real society. At the same time, the effectiveness of some control methods in the introduction part has been verified by controlling the inverted pendulum system. Therefore, it has important significance for the research of inverted pendulum. In order to study the signal tracking problem, we consider the inverted pendulum model based on PID and RBF neural network control. The inverted pendulum model is similar to [48]. The force analysis of the inverted pendulum system is shown in Figure 1.Figure 1
Force analysis diagram of trolley and swing rod.
(a)(b)In Figure1, the external force exerted on the trolley and its moving displacement are indicated by symbols u and x, respectively. θ is expressed as the angle between the pendulum and the vertical direction. After differentiating its displacement is x˙ that is the velocity of the trolley, and the friction coefficient b between it and the trolley and the guide rail phase. Then we can obtain the resistance bx˙ of the guide rail to the trolley in the horizontal direction.In addition, the interaction between the trolley and the pendulum is decomposed into two forces perpendicular to each other in the vertical plane, wherefV and fH are, respectively, used to represent the component force in the vertical and horizontal directions.We use three motions to express the pendulum bar motion of an inverted pendulum: the horizontal motion of the center of gravity, the vertical motion of the center of gravity, and the rotation around the center of gravity. According to Newton's law of mechanics, we can get three equations of motion,ϕ=θ+π represents the angle between the pendulum and the vertical downward direction:(1)fH=md2dt2x+lsinϕ,(2)fV−mg=md2dt2lcosϕ,(3)−fVlsinϕ−fHlcosϕ=Id2ϕdt2.Equation (1) is equivalent to the following equation:(4)fH=mx¨+mlϕ¨cosϕ−mlϕ˙2sinϕ.Equation (2) is also equivalent to the following equation:(5)fV−mg=−mlϕ¨sinϕ−mlϕ˙2cosϕ.The resultant force in the horizontal direction of the trolley can be expressed as follows:(6)Mx¨=u−bx˙−fH.Put (4) into (6), the external force u can be written as(7)u=M+mx¨+mlϕ¨cosϕ−mlϕ˙2sinϕ+bx˙.Similarly, substituting (4) and (5) into (3)yields(8)−mlx¨cosϕ=I+ml2ϕ¨+mglsinϕ.Formulas (7) and (8) are nonlinear equations of motion of vehicle-mounted inverted pendulum system. In order to facilitate the control, linearize the system. Suppose that θ≤20° is within the error range of keeping stability. Because ϕ=θ+π, θis so small; therefore, cosϕ≈−1, sinϕ=θ, and ϕ˙2=θ˙2≈0. After linearization, the system is transformed into the following mathematical model:(9)M+mx¨+bx˙−mlθ¨=u,I+ml2θ¨−mglθ=mlx¨.Taking the Laplace transform in (9), one obtains(10)M+mXss2+bXss−mlΦss2=Us,I+ml2Φss2−mglΦs=mlXss2.By eliminatingXs from the equation set (10), the transfer function of the trolley to the pendulum angle is obtained as follows:(11)Gs=ΦsUs=ml/qs2s4+bI+ml2/qs3−M+mmgl/qs2−bmgl/qs,where q=M+mI+ml2−ml2 is a constant.


## 3. Neural Network Supervision Control Design

### 3.1. PID-RBF and IPID-RBF Control Design

Here, we introduce the PID-RBF control and IPID-RBF control. As we know, PID controller consists of three important parameters, which are proportional regulation coefficientkp, integral regulation coefficient ki, and differential regulation coefficient kd. The proportional regulation coefficient kp can change the response speed of the system and improve the regulation precision of the system. The integral adjustment coefficient ki can eliminate the residual error. The dynamic performance of the system can be improved by differential adjustment coefficient kd. As shown in Figure 2, different PID parameters have different response speeds and stability. When the response curve oscillates significantly, kp should be increased, ki should be increased, and kd should be smaller. When the error of the response curve is large, kp should be reduced, ki should be reduced, and kd should be increased appropriately. According to this method, the best parameters are selected to achieve the best control effect of the inverted pendulum system.Figure 2
Step response curves under different PID parameters.RBF neural network is a three-layer feed-forward neural network, and the mapping from input to output is linear which greatly speeds up the learning speed and avoids the problem of local minima. RBF neural network supervision control is to study the traditional controller, adjust the weight of the network online, and make the feedback control inputupk tend to zero. The structure of the RBF neural network supervision control system is shown in Figure 3.Figure 3
Structure of neural network supervision control system.In the RBF network structure, the input signal of the network is taken asrk, H=h1,…,hmT is radial basis function vector, and Gaussian basis function is hj which is expressed as follows:(12)hj=exp−rk−Cj22bj2,where j=1,…,m, bjis the base width parameter of node j, bj>0, Cj is the center vector of node j, Cj=c11,…,c1mT, and bj=b1,…,bmT.The weight vector of the network is given by(13)W=w1,…,wmT.The output of RBF network is denoted by(14)unk=h1w1+⋯hjwj+⋯+hmwm,where m is the number of hidden layer neurons in the network.The control law is given by(15)uk=upk+unk.The performance indicators of the neural network adjustment are given by(16)Ek=12unk−uk2.The approximation is as follows:(17)∂upk∂wjk≈∂unk∂wik.The error caused by the approximation is compensated by weight adjustment. The gradient descent method is adopted to adjust the weights of the network.(18)Δwjk=−ηunk−ukhjk=−η∂Ek∂wjkhjk,where η is the learning rate.α is the momentum factor, and we get the adjustment process of neural network weights as follows:(19)Wk=Wk−1+ΔWk+αWk−1−Wk−2.(1)
PID controller based on the RBF neural network supervision control method (PID-RBF) includes error signale1k, cumulative error signal ∑s=0ke1s, current error, and last time error difference signal Δe1k=e1k−e1k−1 processing.The PID-RBF control law can be expressed as follows:(20)u¯pk=kp∗e1k+ki∗T1∗∑s=0ke1s+kdT1∗e1k−e1k−1.The network input vector of the PID-RBF control is given by(21)Xk=x1k,x2k,x3kT=e1k∑s=0ke1se1k−e1k−1,whereT1 represents sampling time, e1k=rk−y1k, the input signal is rk, and y1k is the output sequence of the system response, s=0,1,…,k.The network output of the PID-RBF control is(22)u¯nk=h1w¯1+⋯hjw¯j+⋯+hmw¯m.According to (15), (20), and (22), we can express its total control law as(23)u¯k=u¯pk+u¯nk=kp∗e1k+ki∗T1∗∑s=0ke1s+kdT1∗e1k−e1k−1+h1w¯1+⋯hjw¯j+⋯+hmw¯m.The weight adjustment process of RBF neural network supervision is expressed as(24)Δw¯jk=−ηu¯nk−u¯khjk=−η∂E¯k∂w¯jkhjk,w¯jk=w¯jk−1+Δw¯jk+αw¯jk−1−w¯jk−2=w¯jk−1−ηu¯nk−u¯khjk+αw¯jk−1−w¯jk−2.(2)
An improved PID controller based on RBF neural network supervision control strategy (IPID-RBF) includes error signale2k, cumulative error signal e˜2k, current error, and last time error difference signal Δe2k=e2k−e2k−1 processing.(25)u˜pk=kp∗e2k+ki∗T2∗e˜2k+kdT2∗e2k−e2k−1.The network input vector of the IPID-RBF control is(26)X˜1k=x˜1k,x˜2k,x˜3kT=e2ke˜2ke2k−e2k−1,where y2k is the output sequence of the system response, e2k=rk−y2k, e˜2k=e20∗T2+e21∗T2+⋯+e2k−1∗T2+e2k, and T2 represents sampling time.The network output of the IPID-RBF control is as follows:(27)u˜nk=h1w˜1+⋯hjw˜j+⋯+hmw˜m.According to (15), (25), and (27), we can express its total control law as(28)u˜k=u˜pk+u˜nk=kp∗e2k+ki∗T2∗e˜k+kdT2∗e2k−e2k−1+h1w˜1+⋯hjw˜j+⋯+hmw˜m.The weight adjustment process of RBF neural network supervision is expressed as(29)Δw˜jk=−ηu˜nk−u˜khjk=−η∂E˜k∂w˜jkhjk,w˜jk=w˜jk−1+Δw˜jk+αw˜jk−1−w˜jk−2=w˜jk−1−ηu˜nk−u˜khjk+αw˜jk−1−w˜jk−2.


### 3.2. Control Algorithm Stability Analysis

For the stability analysis of the control algorithm, we fully consider the performance indicators adjusted by the neural networkEk, equation (16) Ek=1/2unk−uk2. The equation of state for the free motion of a linear stationary discrete system is(30)xk+1=Gxk.The discrete Lyapunov equation is(31)GTPG−P=−Q,where G is system matrix and P and Q are positive definite real symmetric matrices.The Lyapunov function is expressed as(32)Vx=xTPx.WhenV0=0, x is the solution of the following state equation [47]:(33)xk+1=x′kPxk=Gxk.The increment of the Lyapunov function is(34)ΔVxk=Vxk+1−Vxk.The PID-RBF control functionV1xk=1/2∑i=0ke¯12i. From equation (34), ΔV1xk can be expressed as(35)ΔV1xk=V1xk+1−V1xk=12∑i=0k+1e¯12i−∑i=0ke¯12i=12∑i=0ke¯12i+1−e¯12i=12∑i=0ke¯1i+Δe¯1i2−e¯12i=12∑i=0k2e¯1i⋅Δe¯1i+e¯12i.Combining the previous error and weight adjustment methods, we can expresse1k+1 as(36)e¯1k+1=e¯1k+Δe¯1k=e¯1k+∂e¯1k∂w¯k⋅Δw¯k.The gradient descent method is used to adjust the weight of the network which can be rewritten as(37)Δw¯jk=−η∂E¯k∂w¯jkhjk=−η∂E¯k∂e¯1k∂e¯1k∂w¯jkhjk.Combining equations (36) and (37) and performance indicators E¯k=1/2u¯nk−u¯k2=1/2e¯12k, we can get the following equation:(38)Δe¯1k=∂e¯1k∂w¯kT×Δw¯t=∂e¯1k∂w¯kT×−η∂E¯k∂e¯1k∂e¯1k∂w¯khjk=−ηhjkG1G1T·e¯1k,where G1=∂e¯1k/∂w¯k.According to equation (38), (35) can be expressed as follows:(39)ΔV1xk=−12∑i=0kG1T⋅e¯1kT×2ηhj−η2hj2G1G1TG1T⋅e¯1k.When the value field ofηhj is0<ηhj<2G1G1T−1, we can get2ηhj−η2hj2G1G1T>0. We analyse equation (39) and find that the product of G1T·e¯1kT and G1T·e¯1k is positive. Therefore, we can conclude ΔV1xk<0. When ΔV1xk<0, the system is stable. Because of V1xk=1/2∑i=0ke¯12i, the relation between the preceding and the following is expressed as 1/2e¯12k+1<1/2e¯12k, and then e¯1k+1<e¯1k, limk⟶∞e¯1k=0. Therefore, the control algorithm is convergent.


## 3.1. PID-RBF and IPID-RBF Control Design

Here, we introduce the PID-RBF control and IPID-RBF control. As we know, PID controller consists of three important parameters, which are proportional regulation coefficientkp, integral regulation coefficient ki, and differential regulation coefficient kd. The proportional regulation coefficient kp can change the response speed of the system and improve the regulation precision of the system. The integral adjustment coefficient ki can eliminate the residual error. The dynamic performance of the system can be improved by differential adjustment coefficient kd. As shown in Figure 2, different PID parameters have different response speeds and stability. When the response curve oscillates significantly, kp should be increased, ki should be increased, and kd should be smaller. When the error of the response curve is large, kp should be reduced, ki should be reduced, and kd should be increased appropriately. According to this method, the best parameters are selected to achieve the best control effect of the inverted pendulum system.Figure 2
Step response curves under different PID parameters.RBF neural network is a three-layer feed-forward neural network, and the mapping from input to output is linear which greatly speeds up the learning speed and avoids the problem of local minima. RBF neural network supervision control is to study the traditional controller, adjust the weight of the network online, and make the feedback control inputupk tend to zero. The structure of the RBF neural network supervision control system is shown in Figure 3.Figure 3
Structure of neural network supervision control system.In the RBF network structure, the input signal of the network is taken asrk, H=h1,…,hmT is radial basis function vector, and Gaussian basis function is hj which is expressed as follows:(12)hj=exp−rk−Cj22bj2,where j=1,…,m, bjis the base width parameter of node j, bj>0, Cj is the center vector of node j, Cj=c11,…,c1mT, and bj=b1,…,bmT.The weight vector of the network is given by(13)W=w1,…,wmT.The output of RBF network is denoted by(14)unk=h1w1+⋯hjwj+⋯+hmwm,where m is the number of hidden layer neurons in the network.The control law is given by(15)uk=upk+unk.The performance indicators of the neural network adjustment are given by(16)Ek=12unk−uk2.The approximation is as follows:(17)∂upk∂wjk≈∂unk∂wik.The error caused by the approximation is compensated by weight adjustment. The gradient descent method is adopted to adjust the weights of the network.(18)Δwjk=−ηunk−ukhjk=−η∂Ek∂wjkhjk,where η is the learning rate.α is the momentum factor, and we get the adjustment process of neural network weights as follows:(19)Wk=Wk−1+ΔWk+αWk−1−Wk−2.(1)
PID controller based on the RBF neural network supervision control method (PID-RBF) includes error signale1k, cumulative error signal ∑s=0ke1s, current error, and last time error difference signal Δe1k=e1k−e1k−1 processing.The PID-RBF control law can be expressed as follows:(20)u¯pk=kp∗e1k+ki∗T1∗∑s=0ke1s+kdT1∗e1k−e1k−1.The network input vector of the PID-RBF control is given by(21)Xk=x1k,x2k,x3kT=e1k∑s=0ke1se1k−e1k−1,whereT1 represents sampling time, e1k=rk−y1k, the input signal is rk, and y1k is the output sequence of the system response, s=0,1,…,k.The network output of the PID-RBF control is(22)u¯nk=h1w¯1+⋯hjw¯j+⋯+hmw¯m.According to (15), (20), and (22), we can express its total control law as(23)u¯k=u¯pk+u¯nk=kp∗e1k+ki∗T1∗∑s=0ke1s+kdT1∗e1k−e1k−1+h1w¯1+⋯hjw¯j+⋯+hmw¯m.The weight adjustment process of RBF neural network supervision is expressed as(24)Δw¯jk=−ηu¯nk−u¯khjk=−η∂E¯k∂w¯jkhjk,w¯jk=w¯jk−1+Δw¯jk+αw¯jk−1−w¯jk−2=w¯jk−1−ηu¯nk−u¯khjk+αw¯jk−1−w¯jk−2.(2)
An improved PID controller based on RBF neural network supervision control strategy (IPID-RBF) includes error signale2k, cumulative error signal e˜2k, current error, and last time error difference signal Δe2k=e2k−e2k−1 processing.(25)u˜pk=kp∗e2k+ki∗T2∗e˜2k+kdT2∗e2k−e2k−1.The network input vector of the IPID-RBF control is(26)X˜1k=x˜1k,x˜2k,x˜3kT=e2ke˜2ke2k−e2k−1,where y2k is the output sequence of the system response, e2k=rk−y2k, e˜2k=e20∗T2+e21∗T2+⋯+e2k−1∗T2+e2k, and T2 represents sampling time.The network output of the IPID-RBF control is as follows:(27)u˜nk=h1w˜1+⋯hjw˜j+⋯+hmw˜m.According to (15), (25), and (27), we can express its total control law as(28)u˜k=u˜pk+u˜nk=kp∗e2k+ki∗T2∗e˜k+kdT2∗e2k−e2k−1+h1w˜1+⋯hjw˜j+⋯+hmw˜m.The weight adjustment process of RBF neural network supervision is expressed as(29)Δw˜jk=−ηu˜nk−u˜khjk=−η∂E˜k∂w˜jkhjk,w˜jk=w˜jk−1+Δw˜jk+αw˜jk−1−w˜jk−2=w˜jk−1−ηu˜nk−u˜khjk+αw˜jk−1−w˜jk−2.


## 3.2. Control Algorithm Stability Analysis

For the stability analysis of the control algorithm, we fully consider the performance indicators adjusted by the neural networkEk, equation (16) Ek=1/2unk−uk2. The equation of state for the free motion of a linear stationary discrete system is(30)xk+1=Gxk.The discrete Lyapunov equation is(31)GTPG−P=−Q,where G is system matrix and P and Q are positive definite real symmetric matrices.The Lyapunov function is expressed as(32)Vx=xTPx.WhenV0=0, x is the solution of the following state equation [47]:(33)xk+1=x′kPxk=Gxk.The increment of the Lyapunov function is(34)ΔVxk=Vxk+1−Vxk.The PID-RBF control functionV1xk=1/2∑i=0ke¯12i. From equation (34), ΔV1xk can be expressed as(35)ΔV1xk=V1xk+1−V1xk=12∑i=0k+1e¯12i−∑i=0ke¯12i=12∑i=0ke¯12i+1−e¯12i=12∑i=0ke¯1i+Δe¯1i2−e¯12i=12∑i=0k2e¯1i⋅Δe¯1i+e¯12i.Combining the previous error and weight adjustment methods, we can expresse1k+1 as(36)e¯1k+1=e¯1k+Δe¯1k=e¯1k+∂e¯1k∂w¯k⋅Δw¯k.The gradient descent method is used to adjust the weight of the network which can be rewritten as(37)Δw¯jk=−η∂E¯k∂w¯jkhjk=−η∂E¯k∂e¯1k∂e¯1k∂w¯jkhjk.Combining equations (36) and (37) and performance indicators E¯k=1/2u¯nk−u¯k2=1/2e¯12k, we can get the following equation:(38)Δe¯1k=∂e¯1k∂w¯kT×Δw¯t=∂e¯1k∂w¯kT×−η∂E¯k∂e¯1k∂e¯1k∂w¯khjk=−ηhjkG1G1T·e¯1k,where G1=∂e¯1k/∂w¯k.According to equation (38), (35) can be expressed as follows:(39)ΔV1xk=−12∑i=0kG1T⋅e¯1kT×2ηhj−η2hj2G1G1TG1T⋅e¯1k.When the value field ofηhj is0<ηhj<2G1G1T−1, we can get2ηhj−η2hj2G1G1T>0. We analyse equation (39) and find that the product of G1T·e¯1kT and G1T·e¯1k is positive. Therefore, we can conclude ΔV1xk<0. When ΔV1xk<0, the system is stable. Because of V1xk=1/2∑i=0ke¯12i, the relation between the preceding and the following is expressed as 1/2e¯12k+1<1/2e¯12k, and then e¯1k+1<e¯1k, limk⟶∞e¯1k=0. Therefore, the control algorithm is convergent.


## 4. Simulation Studies

This section provides some simulations to show the inverted pendulum tracking effect of PID-RBF supervision control and IPID-RBF supervision control. We consider the inverted pendulum model and take the swing angle of the swing rod as the controlled object. Under zero initial conditions,M=1.096, m=0.109, l=0.25, g=9.8, I=0.0034, and b=0.1. The transfer function can be expressed as(40)Gs=2.3566s2s4+0.0883s3−27.8285s3−2.3094s.The transfer function of the inverted pendulum system is discretized byz transformation. The discretized object after z transformation is(41)yk=−den2∗yk−1−den3∗yk−2−den4∗yk−3+num2∗uk−1+num3∗uk−2+num4∗uk−3.The base width parameter is given by(42)c=−2,−1,1,2T.The center vector is as follows:(43)b=0.5,0.5,0.5,0.5T.We considerkp=300, kd=100, ki=500, ts=0.001, η=0.30, and α=0.05.In Figure4, the chart is given a square wave signal using neural network supervision control and traditional PID control. Its amplitude is one. From the diagram, it can be seen that the amplitude oscillation of the neural network supervision control is smaller than that of the traditional PID controller. The IPID-RBF supervision control tends to be stable, fastest, and more gentle. Obviously, PID-RBF and IPID-RBF supervision control have stable speed and high accuracy compared with pure PID control.Figure 4
Square wave tracking.In Figure5, the graph shows the RBF neural network supervision control tracking the input square wave signal parameter curve, the un is the RBF network supervision control online learning adjustment curve, and the up is the PID adjustment curve in the RBF network supervision control; superposed curves u are the sum of un and up in RBF networks supervised control. Comparing two adjustments of three curves, because of the transformation of square wave signal from positive to negative, the value of RBF network supervision control online learning adjusting curve is changed from zero to positive, the PID adjusting curve in RBF network supervision control is negative at the jump instant, and the variation of the superposition curve is relatively smooth.Figure 5
Square wave tracking RBF control parameter.
(a)(b)(c)In Figure6, the diagram is given step signal using RBF neural network supervision control and traditional PID control. Its amplitude is one. From the diagram, the amplitude oscillation of the RBF network supervisory control is smaller than that of the conventional PID control. PID-RBF supervision control is stable after 3.8 s. IPID-RBF supervision control is stable after 2.5 s. Obviously, IPID-RBF supervision control is more stable and accurate.Figure 6
Step signal tracking.In Figure7, the graph shows the RBF neural network supervision control tracking the input parameter curve of the step signal. RBF neural network supervision control online learning adjustment curve un and superposition curve uchange trend are basically consistent.Figure 7
Square wave tracking RBF control parameter.
(a)(b)(c)In Figure8, the graph is given sine wave signal r using RBF neural network supervision control and traditional PID control to track. Its amplitude is one. As we can see from the picture, the amplitude oscillation of the RBF network supervision control is smaller than that of the pure PID, and the effect of the RBF neural network supervision control is better than that of pure PID control in the time period from 0s to 20 s. There is no error coincidence between the RBF neural network supervision control curve (y1ory2) and the r curve. The RBF network supervision control has less error and better accuracy than the pure PID control. It can be clearly observed from the partially enlarged view that the IPID-RBF control has the highest coincidence and the best tracking effect.Figure 8
Sine wave tracking.In Figure9, the diagram shows the RBF neural network supervision control tracking the parameter curve of the input sine wave signal. In RBF network supervision control, the change of PID regulating curve up is small, and the change of u in RBF network supervision control is gentle.Figure 9
Sine wave tracking RBF control parameter
(a)(b)(c)From Figures8 and 9, the RBF neural network supervision control online learning adjustment curve un and the input signal r show the opposite trend change under the input sine wave signal. As the input signal increases, the adjustment curve un decreases. The input signal r decreases, and the adjustment curve un increases. The adjustment trend of un is related to the change of weight w. In the process of adjustment, the value of h is positive, un=∑j=1j=mhjwj. When the value of the input signal is positive, the weight w changes ∑j=1j=mhjwj correspondingly and the value is negative; when the value of the input signal is negative, the weight w changes ∑j=1j=mhjwj correspondingly and the value is positive.Figure10 shows that, for a given step signal with an amplitude of one, a given pulse-type disturbance amplitude is one. Interference time is 0.5 s. The pure PID and the RBF neural network supervision control are used for tracking, respectively. The disturbance is added when the time is 5 s in Figure 10. And the amplitude oscillation of the RBF network supervision control is smaller than that of the pure PID control when the disturbance appears. Then, the system can adjust to the stable state quickly after the disturbance disappears to realize the step signal tracking.Figure 10
Step signal tracking.Figure11 shows the RBF neural network supervision control with known disturbance signal and step signal input. There are three large-scale adjustments in total: the adjustment of the system stability before the disturbance is added, the adjustment of the error caused by the disturbance compensation, and the readjustment of the system stability. In RBF network supervision control, the curves un and u tend to be stable after online learning.Figure 11
Step signal tracking RBF control parameter.
(a)(b)(c)In Figure12, the graph is given a square wave signal. Its signal amplitude is one. Given pulse-type disturbance amplitude is two, and the time duration is 0.5 s. When time is 5 s, the disturbance is added. When the disturbance appears, the amplitude oscillation of RBF network supervision control is smaller than that of the pure PID. It is observed in the picture that the system quickly adjusts to the stable state without oscillation and realizes the square wave signal tracking under the IPID-RBF control.Figure 12
Square wave tracking.In Figure13, the picture shows the RBF neural network supervision control with the input of square wave signal and known disturbance signal. In the diagram, there are three large-scale adjustments between 0 and 10 seconds, such as the adjustment of system stability before the disturbance is added, the adjustment of the error caused by disturbance compensation, and the adjustment of system stability. Combined with the regulation of compensating disturbance in the case of step input, traditional PID and the RBF neural network supervision control are used in the regulation of compensating disturbance for the time of 0.5 s. After 0.5 s, the stability of the system is adjusted to stable tracking signal. Among them, the IPID-RBF control had the least amount of speed to adjust.Figure 13
Square wave tracking RBF control parameter.
(a)(b)(c)In Figure14, the graph is given sine wave signal. Its signal amplitude is one. Given pulse-type disturbance amplitude is two, and the time duration is 0.5 s. When time is 5 s, the disturbance is added. It can be seen that the deviation of y1 or y2 from r is less than that of the pure PID, and the curve of RBF neural network supervision control has a higher coincidence with the given sine signal after the disturbance disappears.Figure 14
Sine wave tracking.Figure15 shows that the disturbance amplitude is given to one, and the time duration is 0.5 s, which is monitored by RBF network with the input of sine wave. Compared with the unperturbed condition, the value of the online learning adjustment curve un of RBF neural network supervision control is larger. There are three large pulse-type jumps of un and up superposed curves u in RBF network supervisory control. The other time RBF network supervision control online learning adjustment curve un and the RBF neural network supervision control superposition curve u change trend are basically consistent.Figure 15
Sine wave tracking RBF control parameter.
(a)(b)(c)Combining Table1 and the error graphs from Figures 16 to 21 , we can find that the traditional PID control has the worst effect, long adjustment time, and greater error than the other two control algorithms. The IPID-RBF control can reach a stable state in a short time no matter whether there is no interference or interference.Table 1
The values of error whenk = 20000.
Serial numberDisturbanceInput signale (k)e1 (k)e2 (k)(1)InexistenceStep signal−0.0020−1.6054e − 130.0013(2)ExistenceStep signal−0.00202.4204e − 100.0013(3)InexistenceSine signal−0.02028.2377e − 48.6502e − 4(4)ExistenceSine signal−0.02027.9602e − 48.4492e − 4(5)InexistenceSquare signal0.00400.0014−0.0013(6)ExistenceSquare signal0.00400.0014−0.0014Figure 16
Step signal tracking error without interference.Figure 17
Step signal tracking error under interference.Figure 18
Sinusoidal signal tracking error without interference.Figure 19
Sinusoidal signal tracking error under interference.Figure 20
Wave signal tracking error without interference.Figure 21
Tracking error of wave signal under interference.


## 5. Conclusions

From the simulation effect, the IPID-RBF control applied to the model has better performance. At the same time, the input signal tracking is well achieved. Under three different input signals, real-time tracking of the input signal is realized through online learning, and the error of the input and output is continuously adjusted, so that the system error eventually approaches zero. Compared with the traditional PID control, the IPID-RBF control has the best tracking effect with the input signal under three different input signals. It has improved the characteristics of traditional PID with low accuracy. Compared with the PID-RBF supervision control, the IPID-RBF control has smaller curve oscillation amplitude during the adjustment process, and the system reaches a stable state in a short time and has strong anti-interference ability. This algorithm has simple control and good tracking accuracy.Therefore, the improved control algorithms have good robustness, and the stability of the system is good. Simulation graphics and data show that the IPID-RBF controls the controlled object through online learning to achieve online identification and control. It has high control accuracy, good dynamic characteristics, and anti-interference ability.



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*Source: 5536573-2021-03-23.xml*