# A Novel Smith Predictive Linear Active Disturbance Rejection Control Strategy for the First-Order Time-Delay Inertial System

Authors: Guangda Chen; Dejun Liu; Yongxin Mu; Jinfei Xu; Yanming Cheng
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The control strategy research of the time-delay system is a focused issue in the control field. In order to furthermore improve the performance of the first-order time-delay inertial system, firstly, a new Smith predictor structure is proposed, which solves the constraint that the conventional Smith predictor needs to match the actual object model. Secondly, the performance and parameter function of the new Smith predictor are discussed in theory to provide the basis for parameter tuning. Finally, a new Smith predictor combined with linear active disturbance rejection control (LADRC) is proposed to solve the problem that the two input signals of the linear extended state observer (LESO) are not synchronized on the time scale, and the stability of the new Smith + LADRC time-delay control system is proved theoretically for known and unknown controlled complex objects. Simulation analysis is conducted to verify the robustness of the proposed strategy under the condition of the different parameters. The results indicate that the proposed strategy has better performance than the conventional method in response speed, overshoot, adjustment time, and stability.


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## Body

## 1. Introduction

Time-delay system widely exists in the industrial production process, which is a difficult problem of process control [1]. Due to the delay of signal, the control quantity cannot be adjusted according to the outputs in time. The longer the time delay is, the more difficult the control is. In addition, the system is time-varying [2]. Improving the control performance of the time-delay system has always been one of the hot spots in the control theory and application fields. The most classical theoretical method is the Smith predictor. However, its prediction ability is greatly affected by disturbance and is sensitive to the variation of parameters of the controlled object. To solve the problem, IBM’s Dahlin proposed Dahlin algorithm by designing a suitably digital controller. Although the lag effect is eliminated, there is a ringing effect. When the model error increases, the control quality decreases, and even the system is unstable. Subsequently, many intelligent controls and composite controls that are composed of several kinds of control appeared, such as adaptive control [3, 4], model-free control [5], neural networks control [6], predictive control [7], Smith + PID [8], PI + multimodel dynamic matrix control [9], and Fractional Order Proportional-Integral-Derivative [10]. These algorithms are suitable for the case that the object has less time delay or variation of parameters.Han proposed ADRC (active disturbance rejection control) method, which transforms the design of complex systems into integrated series systems by the observation and compensation of disturbances. For the time-delay system, Han also proposed ignoring the time-delay method for ADRC, increasing order method for ADRC, predictive output method for ADRC, predictive input method for ADRC, delayed input method for ADRC, and other control methods of ADRC for TDS (time-delay systems) [11]. The ignoring time-delay method for ADRC is sensitive to the change of time constant, and the allowable range of time-delay parameters is narrow. The increasing order method for ADRC leads to the increase of controller parameters. Both the predictive output method for ADRC and the predictive input method for ADRC require identifying or estimating the pure time-delay parameters. The accuracy of identification or estimation directly affects the control performance. The ADRC sliding mode control method is proposed [12], which adopts a smooth nonlinear function to avoid system vibrating. The ADRC time-delay control method based on inverse decoupling control is proposed [13]. The system model needs to be identified. A practical ADRC control method is proposed for the automatic thickness control system of hot continuous rolling by using linear modeling and parameter optimization [14]. Since the extended state observer (ESO) and the error feedback control law of ADRC contain nonlinear functions, it is difficult to analyze the stability, and many parameters need to be tuned. To solve the problem, Gao used linear functions to process the ESO and error feedback control law and proposed the linear ADRC (LADRC) method [15, 16], which makes the stability analysis and parameter tuning easier.A reduced-order ADRC strategy [17] for the time-delay system is proposed based on the characteristics of the small phase delay of the reduced-order observer. In [18], the stability region parameters of LADRC for time-delay systems are obtained by using the double-track method. Furthermore, the combination of LADRC and prediction is one of the effective methods to solve the large time-delay system. In [19], ADRC generalized predictive control (ADRC-GPC) is proposed, which does not need to identify the exact mathematical model of the object online, and greatly reduces the amount of online calculation. In [20], the stability conditions and parameter perturbation of Smith + LADRC for the first-order large time-delay systems are analyzed. In [21], an improved Smith + LADRC strategy is proposed to nonminimum phase time-delay systems, and the stability and robustness of the system are analyzed. In [22], the robustness of Smith + LADRC under the condition that the model has probably been known is investigated.Based on the above investigations, a new Smith predictive LADRC method for the first-order time-delay system is proposed, and its prediction mechanism and stability are analyzed under the condition that the controlled object model is known and roughly known through simulations.


## 2. Principle of LADRC

The active disturbance rejection controller is generally composed of three parts: tracking differentiator (TD), state error control law (SECL), and extended state observer (ESO). Its structure is shown in Figure1. The ESO is the core. TD is the process of arranging the transition, which mainly affects the overshoot of the system. If the demand is moderate for overshoots, the TD will not be used. When the linear function is used in SECL and ESO, the active disturbance rejection is called linear active disturbance rejection, which is beneficial to the theoretical and engineering design and parameter tuning and easy to use the frequency domain method to analyze the stability in engineering.Figure 1
Structure diagram of active disturbance rejection controller.The differential equation of the first-order system is expressed as(1)y˙p=fyp,u,w+bu,where yp is the system output, w is the external disturbance, u is the system input, b is uncertain control gain, and fyp,u,w is an internal synthesis and external disturbance of the system. If the d=fyp,u,w+b−b0u is observed total disturbance (b0 is the estimated value of b), then the first-order system can be expressed as(2)y¨p=d+b0u.Supposex1=yp, x2=d, and then equation (2) can be written as the state space expression(3)x˙=Ax+Bu+Ed˙,y=Cx,where A=0100, B=b00, C=10, E=01.Based on equation (3), a linear extended state observer (LESO) is established, and a Luenberger observer for the second-order system is designed as(4)z˙=Az+Bu+Lyp−y^,y^=Cz,where z=z1z2Tis the state estimation of LESO. L=β1β2T is the observer gain. Control quantity u is written as(5)u=kpr−z1−z2b0,where kp is the gain of the proportional controller. For the first-order system, the parameters of LESO and controller are, respectively, [15] as(6)L=β1β2T=2ω0ω02T,kp=ωc,where ω0 and ωc are the bandwidth of LESO and controller, respectively.


## 3. Proposed Control Strategy for First-Order Time-Delay System

A first-order delay system can be represented as(7)Gs=KTps+1e−τps,where Tp is the time constant. τp is the delay time, and K is the gain of the object. Its differential equation can be expressed as(8)y˙p=−1Tpyp+KTput−τp.Iff=−1/Tpyp+K/Tput−τp−b0u is a total disturbance, then y˙p=fyp,u,d+b0u. It can be determined from the above equation that inputs ypt−τp and ut of LESO are not synchronous in time due to the existence of time delay. To estimate the states of the system effectively, two input signals of LESO need to be processed synchronously. It can realize the synchronization of two input signals of LESO by introducing the Smith predictor, but the conventional Smith LADRC will lead to the instability of the system when the parameter perturbation occurs to the object. For this reason, this paper proposes a new method by combining the Smith predictor with linear active disturbance rejection to form a time-delay system controller with strong robustness. Its structure is depicted in Figure 2. Its simplified structure is shown in Figure 3.Figure 2
Structure diagram of the proposed strategy.Figure 3
Simplified Structure diagram of the proposed strategy.It can be obtained from equations (4) and (6) [20, 21] that(9)z1=ω02+2ω0ss+ω02y+b0ss+ω02w,(10)z2=ω02ss+ω02y−b0ω02s+ω02w.It can be obtained from the structure diagram in the dashed box in Figure2 that(11)yps=Gpe−τps1+Kc+KdsGne−τns1+KAGn1+Kc+KdsGpe−τpsus+Gpe−τps1+Kc+KdsGpe−τpsws.According to Figure2 and equation (11), when yp and yn are inconsistent, the feedback controller Kc+Kds works, which can restrain the disturbance. KA can make the pole far away from the virtual axis or make the unstable pole become the stable pole [21].


### 3.1. The Influence ofKc on the Relationship between Disturbances w and yp

It can be obtained from equation (11) that(12)yps=Gpe−τps1+Kc+KdsGpe−τpsws.It can be determined from equation (12) that the closed-loop characteristic equation is 1+Kc+KdsGpe−τps=0. In order to facilitate the analysis of the time-delay part of the closed-loop characteristic equation, the Pade method is used to approximate e−τs. The approximation can be express as(13)e−τs=2−τs2+τs.Then, the closed-loop characteristic equation ofyps/ws can be written as(14)1+Kc+KdsKTps+12−τps2+τps=0,Tps+1τps+2+Kc+KdsK2−τps=0,Tpτps2+2Tps+τps+2+2KcK−KcKτps+2KKds−2KKdτps2=0,Tpτp−2KKdτps2+2Tp+τp−KcKτp+2KKds+2+2KcK=0.If the closed-loop characteristic equation ofyps/ws is stable, the following conditions must be satisfied:(15)Tpτp−2KKdτp>0,2Tp+τp−KcKτp+2KKd>0,2+2KcK>0.Gne−τns=2/10s+1e−10s is chosen as the object model. The following equation can be obtained:(16)Tpτp−2KKdτp>0⟶10∗10−2∗2∗Kd∗10>0⟹Kd<2.5,2Tp+τp−KcKτp+2KKd>0⟶2∗10+10−Kc∗20+2∗2∗Kd>0⟹15>10Kc−2Kd2+2KcK>0⟶2+2∗2∗Kc>0⟹Kc>−0.5.Figure4 shows the response curve of yp with KA=2, Kc=0.3w=1t for different Kd. Figure 5 shows the response curve of yp with KA=2, Kd=0.2w=1t for different Kc.Figure 4
Response curve ofyp with w=1t for different Kd.Figure 5
Response curve ofyp with w=1t for different Kc.It can be determined that the overshoot decreases with the increase ofKd and the time increases. With the increase of Kc, the times of adjustment increase and the steady-state value decreases. This indicates that the large Kc is beneficial to suppress the disturbance. While the value of Kc increases, the adjustment time increases and the response diverges when Kc is larger than the critical value.


### 3.2. The Influence ofKA on the Relationship between Control Quantity u and yp

WhenGp=Gn, Kd=0.2, Kc=0.3, τp=τn, it can be obtained from equation (10) that(17)yps=Gpe−τps1+KAGnus=Ke−τpsTps+1+KAKus.Whenus=1/s, it can be obtained from the final value Theorem 1 that(18)ess∞=limsypss®0=sKe−τpsTps+1+KAK1s=K1+KAK.Figure6 shows the response curve of yp with Gn=2/10s+1, τn=10, u=1t for different KA. It can be determined that, with the increase of KA, the more the pole moves to the left, the faster the response is. However, with the increase of KA, the output value decreases and the equivalent system gain decreases.Figure 6
Response curve ofyp with u=1t for different KA.


## 3.1. The Influence ofKc on the Relationship between Disturbances w and yp

It can be obtained from equation (11) that(12)yps=Gpe−τps1+Kc+KdsGpe−τpsws.It can be determined from equation (12) that the closed-loop characteristic equation is 1+Kc+KdsGpe−τps=0. In order to facilitate the analysis of the time-delay part of the closed-loop characteristic equation, the Pade method is used to approximate e−τs. The approximation can be express as(13)e−τs=2−τs2+τs.Then, the closed-loop characteristic equation ofyps/ws can be written as(14)1+Kc+KdsKTps+12−τps2+τps=0,Tps+1τps+2+Kc+KdsK2−τps=0,Tpτps2+2Tps+τps+2+2KcK−KcKτps+2KKds−2KKdτps2=0,Tpτp−2KKdτps2+2Tp+τp−KcKτp+2KKds+2+2KcK=0.If the closed-loop characteristic equation ofyps/ws is stable, the following conditions must be satisfied:(15)Tpτp−2KKdτp>0,2Tp+τp−KcKτp+2KKd>0,2+2KcK>0.Gne−τns=2/10s+1e−10s is chosen as the object model. The following equation can be obtained:(16)Tpτp−2KKdτp>0⟶10∗10−2∗2∗Kd∗10>0⟹Kd<2.5,2Tp+τp−KcKτp+2KKd>0⟶2∗10+10−Kc∗20+2∗2∗Kd>0⟹15>10Kc−2Kd2+2KcK>0⟶2+2∗2∗Kc>0⟹Kc>−0.5.Figure4 shows the response curve of yp with KA=2, Kc=0.3w=1t for different Kd. Figure 5 shows the response curve of yp with KA=2, Kd=0.2w=1t for different Kc.Figure 4
Response curve ofyp with w=1t for different Kd.Figure 5
Response curve ofyp with w=1t for different Kc.It can be determined that the overshoot decreases with the increase ofKd and the time increases. With the increase of Kc, the times of adjustment increase and the steady-state value decreases. This indicates that the large Kc is beneficial to suppress the disturbance. While the value of Kc increases, the adjustment time increases and the response diverges when Kc is larger than the critical value.


## 3.2. The Influence ofKA on the Relationship between Control Quantity u and yp

WhenGp=Gn, Kd=0.2, Kc=0.3, τp=τn, it can be obtained from equation (10) that(17)yps=Gpe−τps1+KAGnus=Ke−τpsTps+1+KAKus.Whenus=1/s, it can be obtained from the final value Theorem 1 that(18)ess∞=limsypss®0=sKe−τpsTps+1+KAK1s=K1+KAK.Figure6 shows the response curve of yp with Gn=2/10s+1, τn=10, u=1t for different KA. It can be determined that, with the increase of KA, the more the pole moves to the left, the faster the response is. However, with the increase of KA, the output value decreases and the equivalent system gain decreases.Figure 6
Response curve ofyp with u=1t for different KA.


## 4. The Analysis of Stability and Robustness

### 4.1. The Analysis of Stability

According to Figure2 and equations (5), (6), (9), (10), and (11), the closed-loop structure diagram is shown in Figure 7.where Gm=Gn/1+KAGn1−eτns.Figure 7
The single closed-loop structure diagram of the proposed strategy.(19)G′=11+KAGn1+Gne−τnsKc+KdsGpe−τps1+Kc+KdsGpe−τps,H=kpω02+2kpω0s+ω02skps+ω02,C=kps+ω02⋅b0s2+2ω0+kps+2kpω0+ω02s+kpω02Gm−1,


#### 4.1.1. When the Controlled Object Is Accurately Known

Theorem 1.
When the controlled object is accurately known (KA>0, Kd<2.5, Kc>−0.5, 15>10Kc−2Kd), and the model parameters of the Smith predictor are consistent with the object parameters, the stable condition of the system depends on kp, ω0 and b0.
Prove: According to Figure7, the closed-loop transfer function is(20)Gc=CG′1+CG′H=1/1+KAGn1+Gne−τnsKc+KdsGpe−τps/1+Kc+KdsGpe−τpskps+ω02/b0s2+2ω0+kps+2kpω0+ω02s+kpω02Gm1+1/1+KAGn1+Gne−τnsKc+KdsGpe−τps/1+Kc+KdsGpe−τpskps+ω02/b0s2+2ω0+kps+2kpω0+ω02s+kpω02Gm⋅kpω02+2kpω0s+ω02s/kps+ω02.
When the model parameters of the Smith predictor are consistent with the object parameters (K=Kn, τp=τn, Tp=Tn),(21)Gc=Kne−τpskps+ω02b0s2+2ω0+kpsTns+1+2kpω0+ω02s+kpω02Kn.
Then, the closed-loop characteristic equation of the system is
Ds=b0Tns3+b0+b0Tn2ω0+kps2+b02ω0+kps+Kn2kpω0+ω02s+Knkpω02 Let a3=b0Tn, a2=b0+b0Tn2ω0+kp, a1=b02ω0+kp+Kn2kpω0+ω02, a0=Knkpω02. According to Routh stability criterion, a3, a2, a1, a0>0 and the system is stable when a1a2>a0a3:(22)a1a2−a0a3=b02b0ω0+b0kp+2Knkpω01+2Tnω0+Tnkp+Knω021+2Tnω0>0.


#### 4.1.2. When Object Model Parameters Are Unknown

Supposing the nominal model of the object isGb=Gne−τns, then the actual object is Gs=Gb1+δG. δG is the perturbation of the nominal model and stratifies δGjω≤δGω¯. δGω¯ is the bounded uncertainty of the multiplicative norm.From the closed-loop characteristic equation1+CHG′=0, (23)1+CH11+KAGn1+GbKc+KdsGb1+δG1+Kc+KdsGb1+δG=0.SetGF=Kc+Kds. According to the robust stability criterion, for arbitrary ω , the following equation is stratified:(24)δGω¯<ΔGω=1+KAGb+GF+KAGn+CH+CHGbGFGbGF+KAGFGn+CH+CHGbGFGb.Therefore, it can be determined that the stability and robustness of the system can be verified by selecting an appropriate value ofb0, kp, KA, Kc, Kd, and ω0.


### 4.2. Robustness Analysis of the Proposed Control Strategy

To analyze the robustness of our proposed control strategy in the dashed box of Figure2, Gne−τns=2/10s+1e−10s is supposed as the nominal model of the object and setting Kc=0.5, Kd=0.2, and KA=2. The response curve of yp with u=1t for different τp is shown in Figure 8. It can be determined from the response curve of simulation that the response will overshoot when τp>τn. The larger the value of τp is, the larger the overshoot is, and the longer the adjustment time is. Although the delay time of the system becomes shorter, the rise time increases when τp<τn.Figure 8
Response curve ofyp with u=1t for different τp.The response curve ofyp with u=1t for different Tp is shown in Figure 9. It can be determined from the response curve of simulation that the overshoot decreases, the rise time increases, and the larger the Tp is, the longer the rise time is whenTp>Tn. WhenTp<Tn, the rise time decreases, but the overshoot occurs. The longer the adjustment time is, the smaller the Tp is and the larger the overshoot is.Figure 9
Response curveyp with u=1t for different Tp.The response curve ofyp with u=1t for different K is shown in Figure 10. It can be determined from the response curve of simulation that the overshoot occurs when K>Kn. With K becomes larger, the overshoot becomes larger, the rise time decreases, and adjustment time becomes longer as well as the steady-state value becoming larger. When K<Kn, there is no overshoot, rise time increases, and the steady-state value is small.Figure 10
Response curve ofyp with u=1t for different K.


## 4.1. The Analysis of Stability

According to Figure2 and equations (5), (6), (9), (10), and (11), the closed-loop structure diagram is shown in Figure 7.where Gm=Gn/1+KAGn1−eτns.Figure 7
The single closed-loop structure diagram of the proposed strategy.(19)G′=11+KAGn1+Gne−τnsKc+KdsGpe−τps1+Kc+KdsGpe−τps,H=kpω02+2kpω0s+ω02skps+ω02,C=kps+ω02⋅b0s2+2ω0+kps+2kpω0+ω02s+kpω02Gm−1,


### 4.1.1. When the Controlled Object Is Accurately Known

Theorem 1.
When the controlled object is accurately known (KA>0, Kd<2.5, Kc>−0.5, 15>10Kc−2Kd), and the model parameters of the Smith predictor are consistent with the object parameters, the stable condition of the system depends on kp, ω0 and b0.
Prove: According to Figure7, the closed-loop transfer function is(20)Gc=CG′1+CG′H=1/1+KAGn1+Gne−τnsKc+KdsGpe−τps/1+Kc+KdsGpe−τpskps+ω02/b0s2+2ω0+kps+2kpω0+ω02s+kpω02Gm1+1/1+KAGn1+Gne−τnsKc+KdsGpe−τps/1+Kc+KdsGpe−τpskps+ω02/b0s2+2ω0+kps+2kpω0+ω02s+kpω02Gm⋅kpω02+2kpω0s+ω02s/kps+ω02.
When the model parameters of the Smith predictor are consistent with the object parameters (K=Kn, τp=τn, Tp=Tn),(21)Gc=Kne−τpskps+ω02b0s2+2ω0+kpsTns+1+2kpω0+ω02s+kpω02Kn.
Then, the closed-loop characteristic equation of the system is
Ds=b0Tns3+b0+b0Tn2ω0+kps2+b02ω0+kps+Kn2kpω0+ω02s+Knkpω02 Let a3=b0Tn, a2=b0+b0Tn2ω0+kp, a1=b02ω0+kp+Kn2kpω0+ω02, a0=Knkpω02. According to Routh stability criterion, a3, a2, a1, a0>0 and the system is stable when a1a2>a0a3:(22)a1a2−a0a3=b02b0ω0+b0kp+2Knkpω01+2Tnω0+Tnkp+Knω021+2Tnω0>0.


### 4.1.2. When Object Model Parameters Are Unknown

Supposing the nominal model of the object isGb=Gne−τns, then the actual object is Gs=Gb1+δG. δG is the perturbation of the nominal model and stratifies δGjω≤δGω¯. δGω¯ is the bounded uncertainty of the multiplicative norm.From the closed-loop characteristic equation1+CHG′=0, (23)1+CH11+KAGn1+GbKc+KdsGb1+δG1+Kc+KdsGb1+δG=0.SetGF=Kc+Kds. According to the robust stability criterion, for arbitrary ω , the following equation is stratified:(24)δGω¯<ΔGω=1+KAGb+GF+KAGn+CH+CHGbGFGbGF+KAGFGn+CH+CHGbGFGb.Therefore, it can be determined that the stability and robustness of the system can be verified by selecting an appropriate value ofb0, kp, KA, Kc, Kd, and ω0.


## 4.1.1. When the Controlled Object Is Accurately Known

Theorem 1.
When the controlled object is accurately known (KA>0, Kd<2.5, Kc>−0.5, 15>10Kc−2Kd), and the model parameters of the Smith predictor are consistent with the object parameters, the stable condition of the system depends on kp, ω0 and b0.
Prove: According to Figure7, the closed-loop transfer function is(20)Gc=CG′1+CG′H=1/1+KAGn1+Gne−τnsKc+KdsGpe−τps/1+Kc+KdsGpe−τpskps+ω02/b0s2+2ω0+kps+2kpω0+ω02s+kpω02Gm1+1/1+KAGn1+Gne−τnsKc+KdsGpe−τps/1+Kc+KdsGpe−τpskps+ω02/b0s2+2ω0+kps+2kpω0+ω02s+kpω02Gm⋅kpω02+2kpω0s+ω02s/kps+ω02.
When the model parameters of the Smith predictor are consistent with the object parameters (K=Kn, τp=τn, Tp=Tn),(21)Gc=Kne−τpskps+ω02b0s2+2ω0+kpsTns+1+2kpω0+ω02s+kpω02Kn.
Then, the closed-loop characteristic equation of the system is
Ds=b0Tns3+b0+b0Tn2ω0+kps2+b02ω0+kps+Kn2kpω0+ω02s+Knkpω02 Let a3=b0Tn, a2=b0+b0Tn2ω0+kp, a1=b02ω0+kp+Kn2kpω0+ω02, a0=Knkpω02. According to Routh stability criterion, a3, a2, a1, a0>0 and the system is stable when a1a2>a0a3:(22)a1a2−a0a3=b02b0ω0+b0kp+2Knkpω01+2Tnω0+Tnkp+Knω021+2Tnω0>0.


## 4.1.2. When Object Model Parameters Are Unknown

Supposing the nominal model of the object isGb=Gne−τns, then the actual object is Gs=Gb1+δG. δG is the perturbation of the nominal model and stratifies δGjω≤δGω¯. δGω¯ is the bounded uncertainty of the multiplicative norm.From the closed-loop characteristic equation1+CHG′=0, (23)1+CH11+KAGn1+GbKc+KdsGb1+δG1+Kc+KdsGb1+δG=0.SetGF=Kc+Kds. According to the robust stability criterion, for arbitrary ω , the following equation is stratified:(24)δGω¯<ΔGω=1+KAGb+GF+KAGn+CH+CHGbGFGbGF+KAGFGn+CH+CHGbGFGb.Therefore, it can be determined that the stability and robustness of the system can be verified by selecting an appropriate value ofb0, kp, KA, Kc, Kd, and ω0.


## 4.2. Robustness Analysis of the Proposed Control Strategy

To analyze the robustness of our proposed control strategy in the dashed box of Figure2, Gne−τns=2/10s+1e−10s is supposed as the nominal model of the object and setting Kc=0.5, Kd=0.2, and KA=2. The response curve of yp with u=1t for different τp is shown in Figure 8. It can be determined from the response curve of simulation that the response will overshoot when τp>τn. The larger the value of τp is, the larger the overshoot is, and the longer the adjustment time is. Although the delay time of the system becomes shorter, the rise time increases when τp<τn.Figure 8
Response curve ofyp with u=1t for different τp.The response curve ofyp with u=1t for different Tp is shown in Figure 9. It can be determined from the response curve of simulation that the overshoot decreases, the rise time increases, and the larger the Tp is, the longer the rise time is whenTp>Tn. WhenTp<Tn, the rise time decreases, but the overshoot occurs. The longer the adjustment time is, the smaller the Tp is and the larger the overshoot is.Figure 9
Response curveyp with u=1t for different Tp.The response curve ofyp with u=1t for different K is shown in Figure 10. It can be determined from the response curve of simulation that the overshoot occurs when K>Kn. With K becomes larger, the overshoot becomes larger, the rise time decreases, and adjustment time becomes longer as well as the steady-state value becoming larger. When K<Kn, there is no overshoot, rise time increases, and the steady-state value is small.Figure 10
Response curve ofyp with u=1t for different K.


## 5. Simulations and Discusses

In simulations, the nominal model of a large time-delay system isGb=Gne−τns=2/10s+1e−10s, and parameters of the controller of this nominal model are kp=20, ω0=5, b0=20, KA=2, Kd=0.2, Kc=0.5, rt=1. Figure 11 shows the response curve of square wave disturbance with amplitude 1, frequency of 0.5 rad/sec, andt=100s by using ordinary Smith predictor + LADRC strategy and our proposed strategy (new Smith predictor + LADRC strategy). Figure 12 shows the response curve of the sudden reduction of object gain by 10% when t=100s. Figure 13 shows the response curve of object τp increased by 20%. Figure 14 shows the response curve of the object Tp increased by 10%. Figure 15 shows the response curve when the object gain changes to 0.6kn, 0.8kn, and 1.2kn separately. Figure 16 shows the response curve of object τp  = 6, 12, and 13, respectively. Figure 17 shows the response curve of Tp =  0.8Tn, 1.3Tn, and 1.5Tn.Figure 11
Output response curve with square wave disturbance.Figure 12
Response curve with 10% gain reduction.Figure 13
Response curve of objectτp increased by 20%.Figure 14
Response curve of the objectTp increased by 10%.Figure 15
Response curve ofyp for different K.Figure 16
Response curve ofyp for different τp.Figure 17
Response curve ofyp for different Tp.It can be seen from Figures11–14 that the proposed control strategy has better performance than the ordinary Smith ADRC in suppressing external and internal disturbances. The closed loop is adopted to suppress the decrease of output amplitude that is caused by the introduction of KA. It also can be seen from Figures 15–17 that the new Smith + LADRC has better robustness.In order to verify the effect of the proposed algorithm,G1s=0.85/1200s+1e−1800s in literature [1, 23] is selected as the controlled object. The time-delay parameter τ is larger than the inertia parameter T, which increases the difficulty of control. The controller parameters are selected as KA=0.01, Kd=2, Kc=0.2, respectively.kp, b0 are the same as those in literature [1]. The input signal rt=1, and disturbance signal rt−10000=0.2. The simulation results are compared with literature [1] in tracking performance and anti-interference performance (literature [1] did not do simulation experiments for each parameter change). Comparison of system response curves can be obtained in Figure 18.Figure 18
Response curves of the system when the controlled object isG1s.It can be seen from Figure19 that the control strategy proposed in this paper is superior to the method used in literature [1] in response speed and disturbance rejection.Figure 19
Response curve ofK change.To further verify the adaptability and robustness of the proposed algorithm, the time-delay system with unstable poles is selected, and its transfer function isG2=2/10s−1e−10s. For this system, the parameters of the controller are selected as kp=20, ω0=5, b0=20, KA=3, Kd=3, Kc=0.53. Simulation experiments are conducted under various conditions when input signal rt=1. Figure 20 shows the response curves of the new Smith + LADRC, and Figure 18(b) shows the response curves of the conventional Smith + LADRC. Figure 21 shows the response curve of T change. Figure 22 shows the response curve of τ change. Figure 19 shows the response curve of K change.Figure 20
Response curves of the system when the controlled object isG2s. (a) New smith + LADRC. (b) Conventional smith + LADRC.
(a)(b)Figure 21
Response curve ofT change.Figure 22
Response curve ofτ change.It can be seen from Figures19 and 20 that for the time-delay system with unstable poles, the conventional Smith + LADRC is difficult to achieve stable control, while the new Smith + LADRC proposed in this paper not only can achieve stable control but also has good parameter robustness.


## 6. Conclusions

A new Smith + LADRC strategy is proposed to the first-order large time-delay inertia system. The mechanism of the new Smith predictor and the relationship between parameters and stability of the proposed control strategy are verified via conducting a simulation. Conclusions are as follows:(1)
The performance of the proposed new Smith predictor is not limited by the need to be consistent with the interplant model. It can also be applied to nonminimum phase delay systems.(2)
The robustness of the proposed new Smith predictor is proved by theoretical derivation and numerical simulation. The relationship between parameters and performance indexes is summarized, which can provide guidance for parameters’ tuning.(3)
The proposed new Smith predictor + LADRC strategy solves the problem that the input signal of the extended state observer is not synchronized when the LADRC is applied to the time-delay system and has good robustness to parameter disturbance and external disturbance. This strategy has universal applicability for the control problems of time-delay systems.



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*Source: 5560123-2021-03-25.xml*