# A Novel Active Disturbance Rejection Control with a Super-Twisting Observer for the Rocket Launcher Servo System

Authors: Di-Fen Shi; Yuan-Long Hou; Xiao-Hui Gu; Run-Min Hou
Journal: Shock and Vibration
 (2021)

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## Abstract

In this paper, a novel active disturbance rejection control (NADRC) with a super-twisting extended state observer (SESO) is utilized in the rocket launcher servo system. The main arguments in the shipborne rocket launcher system are control accuracy and antidisturbance ability, which are closely related to phase delay and bandwidth. Firstly, we use Taylor’s formula approach to compensate the phase delay in traditional tracking differentiator (TD). Secondly, we design the parallel structured SESO to improve the observation bandwidth, so that it can estimate states with desired accuracy in NADRC. Finally, sinusoidal simulation results show Taylor’s formula-based TD can suppress noise and compensate phase delays effectively. In comparison with traditional ADRC, the proposed NADRC is shown to have better tracking performance and stronger robustness. Semiphysical experiments are conducted to prove the feasibility of NADRC.


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## Body

## 1. Introduction

Permanent magnet synchronous motors (PMSMs) have many merits such as large torque density, high efficiency, and fast response. Therefore, PMSMs have received significant attention in engineering research field like electric vehicles (EVs), spacecraft, and pneumatic servo system [1]. PMSMs are also used in the shipborne rocket launcher system with advanced control technology [2–6]. The marine condition features multivariable and intricate disturbances, as well as strong nonlinear effects. Therefore, rapidly changing disturbances become the main vulnerability of the shipborne rocket launcher system. In practical working conditions, strong robustness and adaptivity are demanded for a PMSM controller. Various effective control methods are investigated in [7–12].The controller in shipborne rocket launcher system needs to solve the nonlinear issue caused by marine conditions and friction. The proportional-integral-derivative (PID) is an effective scheme for the nonlinear problem. As one of the foremost pioneers in the development of the control method, Jingqing Han proposes the active disturbance rejection control (ADRC) [13] to overcome the shortcomings of the PID; hence, it has been developed in recent years [14]. ADRC commonly consists of a tracking differentiator (TD), an extended state observer (ESO), and a nonlinear state error feedback (NLSEF) system. Numerous techniques are proposed to improve antidisturbance and robustness properties [15, 16].TD based on Taylor’s formula is clarified and analyzed in [17], aiming at tackling the phase-delay issue from TD. The key role of ESO is to observe total disturbances and transfer the real-time observations to NLSEF. Fractional-order extended state observer is presented to observe the total disturbance in [18]. However, high switching gain leads to chattering and steady state fluctuation. A super-twisting extended state observer (SESO) emerges with stability in finite time and has better performance compared with conventional extended state observer in the nonlinear system [19, 20]. Besides, a super-twisting algorithm based higher order sliding mode observer (HOSMO) is devised in [21]. The aforementioned second-order ESO contains the nondifferential term k2signe1, which is hard to reach the second-order sliding mode. By using a HOSMO, this term can successfully converge to the desired sliding surface.Due to the promising property of strong robustness, the sliding mode control has received significant popularity [22–25]. However, the chattering phenomenon is indisputable coupled with control processing and the convergence is also slow. Nonsingular fast terminal sliding mode control (NFTSMC) is recently reported, which can guarantee convergence to zero in finite time and eliminate chattering phenomenon efficiently [26]. Aiming to reduce tracking error and improve antidisturbance property, an adaptive recursive terminal sliding mode (ARTSM) controller is investigated in [27]. The performance of SMC degrades when dealing with chattering problem; several auxiliary control approaches like adaptive fuzzy PID and adaptive neural network have been embedded in FNTSM [28–31]. ADRC with FNTSMC is developed in [32], which retains the merits of the ADRC of antidisturbance and the FNTSMC with fast convergence speed. Moreover, instead of traditional cascade structure, the parallel structure in position and speed loop is one of the most practical ways to improve dynamic responses [33]. The overall structure of shipborne rocket launcher system is shown in Figure 1, where STM32 microcontroller uses C programming language to simulate the position loop and speed loop control. Control computer is used to send command signal through C++ programming language.Figure 1
Structure of the rocket launcher servo system.As the previously mentioned control technologies suffer from some limitations, this paper proposes a NADRC system with a parallel structure, mainly seeking to address the following problems:(1)
Taylor’s formula-based tracking differentiator (TD) is innovated with Taylor’s formula and nonlinear functionfhan⋅. The new method can compensate phase delay and acquire desired control command, which solve the phase-delay problem in traditional TD effectively.(2)
Super-twisting algorithm is used to incorporate with ESO, which forms the so-called SESO. It is applied in NADRC to achieve higher tracking accuracy and better robustness. Utilizing super-twisting algorithm andtanhFale,k3,c function, SESO can estimate disturbance quickly. By raising parameter k3 in tanhFale,k3,c, the observation accuracy and bandwidth can be improved.(3)
Fast nonsingular terminal sliding mode control (FNTSMC) in NADRC is used to replace the traditional nonlinear state error feedback controller. The FNTSMC has the ability to reach the target position in short time with the estimated position, speed, and disturbance by SESO.


## 2. Modeling of PMSM in the Rocket Launcher Servo System

### 2.1. Modeling of PMSM

In this shipborne rocket launcher system, the surface-mounted PMSM is implemented for attitude control. The mathematical model in thed-q coordinate system is given as(1)i˙d=−RLid+pωiq+udL,i˙q=−RLiq−pωid−pψfL+uqL,ω˙=1JKeiq−TL−Bω,where ud, uq are the stator voltages; id, iq are the stator currents; Ld, Lq are the stator inductance; ψf is the magnetic flux; R is the stator resistance; p is the pole-pair number of PMSM; J is the inertia moment of the rotor; B is the friction modulus; ω is the angular velocity of the rotor; Ke is the torque constant (Te=3/2pψfiq=Keiq, p is the pole-pair number); and TL is the load torque.Setting disturbance asξt=−TL/J−Bω/J−Ke/Jiq∗−iq and substituting ξt into the second equation in (1) derives the following equation:(2)ω˙=1JKeiq−TL−Bω,ξt=−TLJ−BωJ−KeJiq∗−iq,ω˙=KeJiq∗+ξt.Definingb=Ke/J as the coefficient of iq∗, we obtain(3)θ˙=ω,ω˙=biq∗+ξt,y=θ.


### 2.2. Overall Structure of the Rocket Launcher System

The rocket launcher servo system is designed as in Figure2. The NADRC controller is constructed with a Taylor’s formula-based tracking differentiator, a super-twisting extended state observer (SESO), and a fast nonsingular terminal sliding mode control (FNTSMC) system.Figure 2
Structure of the PMSM drive in the rocket launcher servo system.


## 2.1. Modeling of PMSM

In this shipborne rocket launcher system, the surface-mounted PMSM is implemented for attitude control. The mathematical model in thed-q coordinate system is given as(1)i˙d=−RLid+pωiq+udL,i˙q=−RLiq−pωid−pψfL+uqL,ω˙=1JKeiq−TL−Bω,where ud, uq are the stator voltages; id, iq are the stator currents; Ld, Lq are the stator inductance; ψf is the magnetic flux; R is the stator resistance; p is the pole-pair number of PMSM; J is the inertia moment of the rotor; B is the friction modulus; ω is the angular velocity of the rotor; Ke is the torque constant (Te=3/2pψfiq=Keiq, p is the pole-pair number); and TL is the load torque.Setting disturbance asξt=−TL/J−Bω/J−Ke/Jiq∗−iq and substituting ξt into the second equation in (1) derives the following equation:(2)ω˙=1JKeiq−TL−Bω,ξt=−TLJ−BωJ−KeJiq∗−iq,ω˙=KeJiq∗+ξt.Definingb=Ke/J as the coefficient of iq∗, we obtain(3)θ˙=ω,ω˙=biq∗+ξt,y=θ.


## 2.2. Overall Structure of the Rocket Launcher System

The rocket launcher servo system is designed as in Figure2. The NADRC controller is constructed with a Taylor’s formula-based tracking differentiator, a super-twisting extended state observer (SESO), and a fast nonsingular terminal sliding mode control (FNTSMC) system.Figure 2
Structure of the PMSM drive in the rocket launcher servo system.


## 3. SESO-Based NADRC

High bandwidth is a critical factor to enhance antidisturbance performance in NADRC. A parallel structure between the position loop and speed loop helps to enlarge the bandwidth and modify the frequency bandwidth, as shown in Figure3, where θ∗t is the desired position, θt is the t position control signal, and ωt is the speed control signal.Figure 3
Parallel structure in position and speed control loop.


### 3.1. Taylor’s Formula-Based Tracking Differentiator

To suppress the chattering of the control input, a tracking differentiator is adopted to achieve smooth tracking. A drawback of a standard TD is that it is not able to deal with phase delay. A traditional TD can be described as(4)θk+1=θk+hωk,ωk+1=ωk+hfhanθk,ωk,r0,h0,where θt is the adjusted position control signal; r is the speed control factor; h0 is the filtering coefficient; and h is the sampling step. Generally, large h0 is conducive to restrain noise and high r can raise tracking accuracy. The nonlinear operator fhan⋅ in equation (4) is commonly used in ADRC applications, which is given in [13](5)d=h0r02,a0=h0ωk,y=θk+a0,a1=dd+8y,a2=a0+signa1−d2,sy=signy+d−signy−d2,a=a0+y−a2sy+a2,sa=signa+d−signa−d2,fhan=−r0ad−signasa−r0signa.The phase delay resulting from TD is compensated by Taylor’s formula approach; Taylor’s formula is used to compensate phase delay in TD [18], and it can be expressed as(6)g˜dkn+1=gdkn+gd′kn1!kn+1−kn+gd″kn2!kn+1−kn2+⋯+gdnknn!kn+1−knn+gdn+1ξn+1!kn+1−knn+1.Denotingθ∗ as the tracking reference and Δt as the time interval, the second-order tracking differentiator is as follows:(7)θ∗t=θt+θ′tΔt+12θ″ξΔt2,where θ∗t is the input and θt+Δt is the output.Theorem 1.
Assumeθi+1t∞<∞, i=0,1. There exists a positive constant a, satisfying(8)limΔt⟶0θit+Δt−θ∗it=0,t∈a.+∞.Proof.
Differential of (7) can be written as(9)θ∗it=θit+θi+1tΔt+12θi+2tΔt2.
Taylor’s formula-based expansion ofθit is(10)θit+Δt=θitΔt+θi+1tΔt+12θi+2ξΔt2,ξ∈t,t+Δt.
The absolute value of equation (10) minus equation (9) is(11)θit+Δt−θ∗it≤12θi+2ξΔt2+12θi+2tΔt2.
Replacingt with s⋅Δt and setting yis=θ∗i+1s⋅Δt, the differential of (8) is(12)θ∗i+1s⋅Δt=yis+ddsyis+12d2ds2yis.
Equation (12) can be rewritten as(13)ddsYis=AYis+ϑ,where(14)Yis=yisddsyis,A=01−2−2,ϑ=02θ∗i+1s⋅Δt.
The solution for equation (13) can be given as(15)Yis=eAsYi0+∫0seAs−τϑτdτ,where A is Hurwitz, σρ>0 and eAsF≤ρe−σs. Then we have(16)Yis2≤eAsYi02+∫0seAs−τϑτ2≤Yi02ρe−σs+∫0sϑτ2ρe−σs−τdτ≤Yi02ρe−σs+2ρσθ∗i+1s⋅Δt∞,(17)12θi+2ξΔt2=12Δt2d2ds2yis=12Δt2Yis2≤ρe−σs2Yi02+ρσθ∗i+1s⋅Δt∞.
Combining1/2θi+2tΔt2⟶0Δt⟶0 and (11), equation (8) is proved. Based on the above analysis, Taylor’s formula-based TD is given as(18)θct+Δt=θt+θ′tΔt+12θ″ξΔt2,ωct+Δt=ωt+Δt⋅fhanθk,ωk,r0,h0+12ω″ξΔt2.


### 3.2. Super-Twisting Extended State Observer

The aforementioned dynamic model of the system can be rewritten as(19)θ˙=ω,ω˙=biq∗+ξt,y=θ.Super-twisting algorithm is applied in SESO, which observesθ, ω, and ξ in (18), respectively. Fal is a nonsmooth function proposed by the researcher Han [13]. Li et al.’s research shows the nonsmooth control has better anti-interference performance than the smooth control [34]. The Fal function is given by(20)Fal=eδ1−α,e≤δ,eαsigne,e>δ.The Fal function is drawn as in Figure4. We note that when 0<α<1, δ is the threshold of nonsmooth interval. Specifically, if e<δ, it is linear which is useful to prevent high-frequency fluctuations caused by high gains in sign function. And if e>δ, it is not smooth. The overall consideration of balance between sign function and linear function leads to constraining the ranges of α and δ within [0 1].Figure 4
Properties of the Fal function with differentα and δ values.
(a)(b)The hyperbolic tangent nonlinear functiontanhFale,β,c is much smoother than the Fal function, which can alleviate chattering problem effectively [35].(21)tanhFale,β,c=βtanhce,where e is the input, and β and c are parameters to be adjusted. tanhFale,β,c function is plotted in Figure 4. It can be seen that if α=0 and e>δ, the output is −β or β. The smaller c results in a smoother transition curve when e is close to 0.The speed error is defined ase=ωct−ω^t. The observed value is expressed as θ^, ω^, and ξ^. The third-order state extended observer is(22)θ^˙=ω^+k1e2/3signe,ω^˙=ξ^+biq∗+k2e1/3signe,ξ^˙=−tanhFale,k3,c.The observer parameters are obtained by settingk1=1.5τ and k2=1.1τ, where the constant boundary τ is defined as ξ^˙<τ. And k3 is time-varying parameter which is determined by β and c, as shown in Figure 5.Figure 5
Properties of thetanhFale,β,c function.


### 3.3. Fast Nonsingular Terminal Sliding Mode Control (FNTSM)

FNTSM approach is proposed for speed control. The speed error is defined ase=ωct−ω^t. Then, the derivative of e is(23)e˙=ω˙ct−ω^˙t=ω˙c−ξ^−biq∗−k2e1/3signe.Nonsingular terminal sliding mode (NTSM) has been proposed for finite-time convergence to zero of the sliding variables.(24)s=e+βsigne˙e˙σ,where β>0 and 1<σ<2.NTSM has the property of finite-time convergence to zero of the sliding variables. Settings=0 yields the relation between position and velocity e=−βsigne˙e˙σ. Note moving from e0 to 0 takes time Tn =∫e001/e˙de. And from (24), we have 1/e˙σ=−βsigne˙/e; thus ∫e001/e˙de=∫0e0βsigne˙/ede. Then we have the convergence time:(25)Tn=∫0e0β1/σe1/σde=β1/σe01−1/σ1−1/σ.NTSM overcomes the singularity problem when the sliding states are far away from the equilibrium condition. Nevertheless, the convergence is slow. To improve convergence speed, FNTSM is proposed according to(26)s=e+βsigne˙e˙σ+k2e1/3signe,where β>0 and 1<σ<2.Similarly, by settings=0, we have e+k2e1/3signe=−βsigne˙e˙σ, which can be 1/e˙σ=βsigne˙/e+k2e1/3signe. And by Tf=∫e001/e˙σde=∫0e0βsigne˙/e+k2e1/3signe, we can derive the time to achieve the equilibrium condition(27)Tf=∫0e0β1/σe+k2e1/31/σde.Note that becausee+k2e1/31/σ>e1/σ, we can prove that(28)Tn=∫0e0β1/σe1/σde>∫0e0β1/σe+k2e1/31/σde=Tf.Therefore, the FNTSM method can achieve the equilibrium point more quickly than the NTSM.For the speed errore, it can reach equilibrium condition s=0 in finite time if FNTSM manifold is given as equation (26), and the corresponding control law is(29)iq∗=b−1ieq+ib−ξ^,(30)ieq=ω˙c+βsigne˙e˙σ,(31)ib=∫0tβ0s+γ0sq/pdτ,where p and q are positive odd numbers p>q, and β0 and γ0γ0>ς are positive constants.According to equations (26) and (29)–(31), we design the following FNTSMC sliding surface:(32)s=e˙ω+βsigne˙e˙σ+k2e1/3signe=ω˙c−biq∗−ξ^−k2e1/3signe+βsigne˙e˙σ+k2e1/3signe.By substituting (29) and (30) into (32), FNTSMC can be derived as follows:(33)s=ω˙c−bb−1ieq+ib−ξ−ξ^−k2e1/3signe+βsigne˙e˙σ+k2e1/3signe=ω˙c−ω˙c+βsigne˙e˙σ+ib−ξ^+βsigne˙e˙σ=−ib−ξ^.Equation (33) is simplified as(34)s=−ib−ξ^.Then, the derivative of (34) is(35)s˙=s=−i˙b−ξ^˙=−β0s−γ0sq/p+tanhFale,k3,c.In order to verify the feasibility of the proposed method and analyze whether speed error can converge to zero in finite time, we set Lyapunov function as(36)V=12s2.It can be calculated as follows:(37)V˙=ss˙=s−β0s−γ0sq/p+tanhFale,k3,c=−β0s2−γ0sq+p/p+tanhFale,k3,cs≤−β0s2−γ0sq+p/p+tanhFale,k3,cs≤−β0s2−γ0−tanhFale,k3,c/sq/psq+p/p≤−β0s2−γ0−ς/sq/psq+p/p=−2β0V−2q+p/2pγ′Vq+p/2p.It can be confirmed thatγ0>ς, which leads to V˙<0. Equation (37) proves equation (26) can be designed as the FNTSM, which converge in finite time.


## 3.1. Taylor’s Formula-Based Tracking Differentiator

To suppress the chattering of the control input, a tracking differentiator is adopted to achieve smooth tracking. A drawback of a standard TD is that it is not able to deal with phase delay. A traditional TD can be described as(4)θk+1=θk+hωk,ωk+1=ωk+hfhanθk,ωk,r0,h0,where θt is the adjusted position control signal; r is the speed control factor; h0 is the filtering coefficient; and h is the sampling step. Generally, large h0 is conducive to restrain noise and high r can raise tracking accuracy. The nonlinear operator fhan⋅ in equation (4) is commonly used in ADRC applications, which is given in [13](5)d=h0r02,a0=h0ωk,y=θk+a0,a1=dd+8y,a2=a0+signa1−d2,sy=signy+d−signy−d2,a=a0+y−a2sy+a2,sa=signa+d−signa−d2,fhan=−r0ad−signasa−r0signa.The phase delay resulting from TD is compensated by Taylor’s formula approach; Taylor’s formula is used to compensate phase delay in TD [18], and it can be expressed as(6)g˜dkn+1=gdkn+gd′kn1!kn+1−kn+gd″kn2!kn+1−kn2+⋯+gdnknn!kn+1−knn+gdn+1ξn+1!kn+1−knn+1.Denotingθ∗ as the tracking reference and Δt as the time interval, the second-order tracking differentiator is as follows:(7)θ∗t=θt+θ′tΔt+12θ″ξΔt2,where θ∗t is the input and θt+Δt is the output.Theorem 1.
Assumeθi+1t∞<∞, i=0,1. There exists a positive constant a, satisfying(8)limΔt⟶0θit+Δt−θ∗it=0,t∈a.+∞.Proof.
Differential of (7) can be written as(9)θ∗it=θit+θi+1tΔt+12θi+2tΔt2.
Taylor’s formula-based expansion ofθit is(10)θit+Δt=θitΔt+θi+1tΔt+12θi+2ξΔt2,ξ∈t,t+Δt.
The absolute value of equation (10) minus equation (9) is(11)θit+Δt−θ∗it≤12θi+2ξΔt2+12θi+2tΔt2.
Replacingt with s⋅Δt and setting yis=θ∗i+1s⋅Δt, the differential of (8) is(12)θ∗i+1s⋅Δt=yis+ddsyis+12d2ds2yis.
Equation (12) can be rewritten as(13)ddsYis=AYis+ϑ,where(14)Yis=yisddsyis,A=01−2−2,ϑ=02θ∗i+1s⋅Δt.
The solution for equation (13) can be given as(15)Yis=eAsYi0+∫0seAs−τϑτdτ,where A is Hurwitz, σρ>0 and eAsF≤ρe−σs. Then we have(16)Yis2≤eAsYi02+∫0seAs−τϑτ2≤Yi02ρe−σs+∫0sϑτ2ρe−σs−τdτ≤Yi02ρe−σs+2ρσθ∗i+1s⋅Δt∞,(17)12θi+2ξΔt2=12Δt2d2ds2yis=12Δt2Yis2≤ρe−σs2Yi02+ρσθ∗i+1s⋅Δt∞.
Combining1/2θi+2tΔt2⟶0Δt⟶0 and (11), equation (8) is proved. Based on the above analysis, Taylor’s formula-based TD is given as(18)θct+Δt=θt+θ′tΔt+12θ″ξΔt2,ωct+Δt=ωt+Δt⋅fhanθk,ωk,r0,h0+12ω″ξΔt2.


## 3.2. Super-Twisting Extended State Observer

The aforementioned dynamic model of the system can be rewritten as(19)θ˙=ω,ω˙=biq∗+ξt,y=θ.Super-twisting algorithm is applied in SESO, which observesθ, ω, and ξ in (18), respectively. Fal is a nonsmooth function proposed by the researcher Han [13]. Li et al.’s research shows the nonsmooth control has better anti-interference performance than the smooth control [34]. The Fal function is given by(20)Fal=eδ1−α,e≤δ,eαsigne,e>δ.The Fal function is drawn as in Figure4. We note that when 0<α<1, δ is the threshold of nonsmooth interval. Specifically, if e<δ, it is linear which is useful to prevent high-frequency fluctuations caused by high gains in sign function. And if e>δ, it is not smooth. The overall consideration of balance between sign function and linear function leads to constraining the ranges of α and δ within [0 1].Figure 4
Properties of the Fal function with differentα and δ values.
(a)(b)The hyperbolic tangent nonlinear functiontanhFale,β,c is much smoother than the Fal function, which can alleviate chattering problem effectively [35].(21)tanhFale,β,c=βtanhce,where e is the input, and β and c are parameters to be adjusted. tanhFale,β,c function is plotted in Figure 4. It can be seen that if α=0 and e>δ, the output is −β or β. The smaller c results in a smoother transition curve when e is close to 0.The speed error is defined ase=ωct−ω^t. The observed value is expressed as θ^, ω^, and ξ^. The third-order state extended observer is(22)θ^˙=ω^+k1e2/3signe,ω^˙=ξ^+biq∗+k2e1/3signe,ξ^˙=−tanhFale,k3,c.The observer parameters are obtained by settingk1=1.5τ and k2=1.1τ, where the constant boundary τ is defined as ξ^˙<τ. And k3 is time-varying parameter which is determined by β and c, as shown in Figure 5.Figure 5
Properties of thetanhFale,β,c function.


## 3.3. Fast Nonsingular Terminal Sliding Mode Control (FNTSM)

FNTSM approach is proposed for speed control. The speed error is defined ase=ωct−ω^t. Then, the derivative of e is(23)e˙=ω˙ct−ω^˙t=ω˙c−ξ^−biq∗−k2e1/3signe.Nonsingular terminal sliding mode (NTSM) has been proposed for finite-time convergence to zero of the sliding variables.(24)s=e+βsigne˙e˙σ,where β>0 and 1<σ<2.NTSM has the property of finite-time convergence to zero of the sliding variables. Settings=0 yields the relation between position and velocity e=−βsigne˙e˙σ. Note moving from e0 to 0 takes time Tn =∫e001/e˙de. And from (24), we have 1/e˙σ=−βsigne˙/e; thus ∫e001/e˙de=∫0e0βsigne˙/ede. Then we have the convergence time:(25)Tn=∫0e0β1/σe1/σde=β1/σe01−1/σ1−1/σ.NTSM overcomes the singularity problem when the sliding states are far away from the equilibrium condition. Nevertheless, the convergence is slow. To improve convergence speed, FNTSM is proposed according to(26)s=e+βsigne˙e˙σ+k2e1/3signe,where β>0 and 1<σ<2.Similarly, by settings=0, we have e+k2e1/3signe=−βsigne˙e˙σ, which can be 1/e˙σ=βsigne˙/e+k2e1/3signe. And by Tf=∫e001/e˙σde=∫0e0βsigne˙/e+k2e1/3signe, we can derive the time to achieve the equilibrium condition(27)Tf=∫0e0β1/σe+k2e1/31/σde.Note that becausee+k2e1/31/σ>e1/σ, we can prove that(28)Tn=∫0e0β1/σe1/σde>∫0e0β1/σe+k2e1/31/σde=Tf.Therefore, the FNTSM method can achieve the equilibrium point more quickly than the NTSM.For the speed errore, it can reach equilibrium condition s=0 in finite time if FNTSM manifold is given as equation (26), and the corresponding control law is(29)iq∗=b−1ieq+ib−ξ^,(30)ieq=ω˙c+βsigne˙e˙σ,(31)ib=∫0tβ0s+γ0sq/pdτ,where p and q are positive odd numbers p>q, and β0 and γ0γ0>ς are positive constants.According to equations (26) and (29)–(31), we design the following FNTSMC sliding surface:(32)s=e˙ω+βsigne˙e˙σ+k2e1/3signe=ω˙c−biq∗−ξ^−k2e1/3signe+βsigne˙e˙σ+k2e1/3signe.By substituting (29) and (30) into (32), FNTSMC can be derived as follows:(33)s=ω˙c−bb−1ieq+ib−ξ−ξ^−k2e1/3signe+βsigne˙e˙σ+k2e1/3signe=ω˙c−ω˙c+βsigne˙e˙σ+ib−ξ^+βsigne˙e˙σ=−ib−ξ^.Equation (33) is simplified as(34)s=−ib−ξ^.Then, the derivative of (34) is(35)s˙=s=−i˙b−ξ^˙=−β0s−γ0sq/p+tanhFale,k3,c.In order to verify the feasibility of the proposed method and analyze whether speed error can converge to zero in finite time, we set Lyapunov function as(36)V=12s2.It can be calculated as follows:(37)V˙=ss˙=s−β0s−γ0sq/p+tanhFale,k3,c=−β0s2−γ0sq+p/p+tanhFale,k3,cs≤−β0s2−γ0sq+p/p+tanhFale,k3,cs≤−β0s2−γ0−tanhFale,k3,c/sq/psq+p/p≤−β0s2−γ0−ς/sq/psq+p/p=−2β0V−2q+p/2pγ′Vq+p/2p.It can be confirmed thatγ0>ς, which leads to V˙<0. Equation (37) proves equation (26) can be designed as the FNTSM, which converge in finite time.


## 4. Simulation Studies

Noise is amplified when using the traditional differentiator with the simultaneous action of the differential operation. With this in view, we attempted to use TD (equation (4)) to reduce noise. Compared with the traditional differentiator, the TD exhibits an antinoise capability, as shown in Figure 6. These experimental results prove the effectiveness of the TD in reducing noise. Certainly, the phase delay occurs in the TD; hence, Taylor’s formula-based TD is carried out to solve this problem. Tracking the differential of y=6⋅sin0.8⋅t, it turns out that Taylor’s formula-based TD remains low phase delay by comparing TD in Figure 7.Figure 6
Differentiator with random noise.Figure 7
Tracking differentiator experiment.In this section, step response and sinusoidal signal tracking experiments are conducted in Matlab. The robustness of the proposed method is validated by the step response with random disturbances experiment, and tracking performance is tested by sinusoidal tracking experiment.It can be learned from Figure8 that NADRC can reach the reference position within 0.75 s, while ADRC and PID use 0.86 s and 1.41 s to reach it. This experimental result shows that , by using FNTSM, NADRC can reach the target position in shorter time. To verify the robustness of the system, step simulation with rapidly changing disturbances is conducted. The simulation results with different control methods are depicted in Figure 8; the proposed NADRC has relatively small fluctuation around 0.11。, while ADRC and PID methods are around 0.13。 and 0.18。, respectively. This simulation shows that NADRC has better antidisturbance ability in working process. It is proved that SESO can improve the antidisturbance practically.Figure 8
Step response simulation.As for the dynamic performance analysis of ADRC, PID, and NADRC, sinusoidal tracking simulation is carried out (see Figure9). It is shown that PID tracking accuracy is worse than ADRC, while the NADRC tracks the reference signal more accurately than the traditional ADRC.Figure 9
Sinusoidal tracking.With sinusoidal tracking error curve drawn in Figure10, the maximum tracking error of NADRC, ADRC, and PID is about 0.034o, 0.086o, and 0.282o, respectively. Compared with PID and ADRC, NADRC has a faster convergence speed. Based on the above results, it can be seen that ADRC tracking error is more than twice that of NADRC, which proves that the proposed NADRC scheme has a faster convergence speed and high tracking accuracy in this sinusoidal tracking simulation. The simulation result is shown in Table 1.Figure 10
Sinusoidal tracking error.Table 1
Parameters in PMSM.
Step simulationSinusoidal trackingControl strategyAdjustment time (s)Maximum error (°)Maximum error (°)PID1.410.180.282ADRC0.860.130.086NADRC0.750.110.034


## 5. Semiphysical Experiment

To further verify the effectiveness of NADRC approach, a semiphysical experiment is conducted. The entire semiphysical experiment platform structure is shown in Figure11. It can be seen that the platform is composed with control computer, sensor system, PA, PRG, AM, RIP, and the support frame. The disturbances are from frictional moment, load torque, and rotational inertia, which are simulated by RIP and MPB in this experiment. The motor parameters are listed in Table 2.Figure 11
Semiphysical platform structure.Table 2
Parameters in PMSM.
ParameterValueLoad moment of inertiaJ5.556×10−3kg×m2Friction coefficientB1.43×10−4N×m/rad×s−1Load disturbing momentTL9.32×103kg×m2Motor torque factorKt0.175N×m/AReduction ratio (i)1039Stator inductanceLd=Lq21.24mHPole pairsp3Motor magnetic fluxψf0.65WbMaximum sampling frequency of D/A transform is100kHz, with power amplifier transfer voltage from 28 V to 36 V, which is 400 Hz. The range of speed measurement is 0.03∼100°/s. Reduction ratio of precision gear box is 360:1. The rotational inertia of plate is 2.315⋅10−4kg⋅m2. The maximum magnetic power of brake is 108.7kgf⋅m. The rated torque is 13.9N⋅m25C,13.1N⋅m40, the maximum torque is 35.3N.M25, the rated power is 5.4kW, the maximum voltage is 250V, the rated current is 19.8A, the maximum current is 40A, and the rated speed is 5000rpm.To verify the dynamic performance of the proposed NADRC scheme, a tracking experiment is conducted on the platform. The trajectory is a sinusoidal signal with a period of 3.768 s and amplitude of30°. The experimental results of position, speed, and acceleration are shown in Figure 12. It can be seen from Figure 12 that the position tracking curve is the smoothest, and acceleration curve has relatively large fluctuation.To further analyze the control accuracy, position tracking curve is partially enlarged in Figure13 and the tracking error is drawn in Figure 14. It can be seen that the maximum position tracking error is hover around 0.6°. As is anticipated, the tracking error of the sinusoidal signal by NADRC meets the dynamic error index 0.2°, which proves its feasibility in practical applications.Figure 12
Sinusoidal tracking curves.
(a)(b)(c)Figure 13
Position tracking curve.Figure 14
Position tracking error curve.


## 6. Conclusion

In this paper, the position and speed control loop is innovated as a parallel structure with the proposed NADRC scheme. Both static and dynamic performances are successfully verified through step response simulation and sinusoidal signal tracking experiments. The step simulations are carried out with rapidly changing random disturbances for comparing PID, ADRC, and NADRC methods. Results show that NADRC can achieve target position quickly with better antidisturbance performance. Sinusoidal simulation results show Taylor’s formula-based TD and SESO can reduce phase delay and acquire precise tracking performance. The position tracking in semiphysical experiment can meet the dynamic error index designed as 0.8°. Based on the above experiments and analysis, the proposed control method can be employed in this rocket launcher servo system, possibly employed in other practical applications in the future.It is interesting that finite-time convergence is in conflict with chattering suppression problem. The super-twisting method has fast convergence speed, but it may cause a larger chattering problem [36]. In our future work, we try to combine it with low switching gain method and Kalman filter to address it.



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*Source: 6617599-2021-06-30.xml*