# Improved Finite Control Set Optimal Control for PMSM in Rocket Launcher Servo System

Authors: Di-fen Shi; Yuan-long Hou; Xiao-hui Gu; Run-min Hou
Journal: Shock and Vibration
 (2020)

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## Abstract

Permanent magnet synchronous motors (PMSMs) have been widely applied in the shipborne rocket launcher systems due to their high torque density and high efficiency. However, since there are many external disturbances from the shipborne rocket launcher, the tracking performance under random noises needs to be improved. In this paper, an improved finite control set optimal control (IFCSOC) based on a super-twisting extended state observer (SESO) is investigated for position tracking control of PMSMs. The SESO can improve the anti-interference ability of the proposed controller. Moreover, in order to improve tracking accuracy, Taylor’s formula is used to solve the phase-lag problem of nonlinear tracking differentiator in IFCSOC. Simulation shows that compared with conventional FCSOC, IFCSOC exhibits better robustness under random disturbances. Furthermore, the semiphysical experiment is conducted to verify the proposed IFCSOC strategy.


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## Body

## 1. Introduction

Permanent magnet synchronous motors (PMSMs) are commonly used in shipborne rocket launcher systems for attitude adjustment distinguished by high torque density and high efficiency. Under the complex marine conditions, the shipborne rocket launcher system is easy to be disturbed and tends to exhibit a nonlinear behavior, which requires the controller to possess strong robustness to the rapidly changing disturbances.Various control methods have been studied for nonlinear systems due to their prominent control performance [1–7]. A sampled-data output feedback controller design technique is investigated in [4], targeting for nonlinear systems with uncontrollable and unobservable linearizations around the origin. Nevertheless, it is not easy to determine whether the wave fluctuation that imposed on the shipborne rocket launcher system satisfies the assumption of such design technique.Finite control set optimal control (FCSOC) is a new control method receiving more and more attention, it can be applied to systems with nonlinearities and predigest control process [5, 6]. However, to gain high performance, the FCSOC strategy needs an accurate model of the system. Considering the application in the shipborne rocket launcher, the rapidly changing disturbances from marine environment will to large extent cause performance degradation. To address this and improve the system robustness, the external disturbances must be well estimated for compensation.Active disturbance rejection control (ADRC) is marked by strong robustness. However, when applied to the high-order process, such design technique and the associated parameter tuning will be complicated. Also, it tends to have the slow convergence speed since it mainly focuses on the low-frequency disturbance [7]. To overcome this and gain better anti-interference ability, sliding-mode active disturbance rejection control (SADRC) is proposed in [7], where the high-order sliding-mode technology is used to construct the estimator for internal coupling factors and external disturbances. In [8], it is combined with a nonlinear extended state observer (NESO) to realize the position loop control of the PMSM. Nevertheless, there still exist two challenges. On the one hand, applying the sliding-mode control technique brings the chattering phenomenon, this requires for estimators with fast convergence algorithms. On the other hand, the phase lag still exists when tracking the reference signals due to the discrete form of nonlinear tracking differentiator (DNTD). To reduce the phase lags, a strict proof for some specific conditions of nonlinear tracking differentiator (NTD) is made by Guo and Zhao [9]. The research studies of the convergence, rapidity improvement and its high-order form application are carried out in [10–13]. Cataloging former references, Jinping Feng proposes a Taylor’s formula-based phase compensation method to decrease phase lags [14].The disturbance estimator is the critical part of ADRC, which is used to compensate the difference between the real model and the nominal model caused by external disturbances and changes of model parameters. There are various disturbance estimators, among which the ESO is one of the most widely used. It is because it requires the least mathematical model information and can describe the internal and external disturbances together [15–17]. The ESO is used to estimate rapidly changing disturbances in [18] and used to estimate the system disturbances for feedforward compensation in the proposed composite continuous fast terminal sliding-mode control (CFTSMC) [19]. For nonlinear systems, the convergence of ESO will be challenged. A practical way is to combine it with the super-twisting algorithm, forming the so-called super-twisting ESO (SESO). The SESO can achieve a better estimation performance as it can achieve finite-time convergence. A longitudinal study of the modified super-twisting algorithm is used to estimate state variables and disturbances in [20]. A SESO is studied in [21] with an adaptive strategy to estimate various disturbances for a diesel engine air path. These research studies show that the SESO exhibits a higher convergence speed than the general ESO for nonlinear systems.This paper proposes the IFCSOC strategy to gain better robustness and tracking accuracy for the PMSM system in the shipborne rocket launcher. In the proposed control strategy, the adaptive disturbance compensation FCSOC strategy is applied for the current loop control, where the SESO is used to achieve finite-time stability. Also, SADRC is combined with Taylor’s formula for the position loop control, reducing the phase lags resulting from the DNTD. Finally, simulation and semiphysical experiments are carried out for validation.


## 2. Model Description

A surface-mounted PMSM is used in this shipborne rocket launcher system. In thed-q coordinate system, the general model is(1)i˙d=−1LdRid+Lqpiqωr+ud+fd,i˙q=−1LqRiq+Ldpidωr−ψfpωr+uq+fqω˙r=1JTe−TL−Bωr,,where ud and uq are stator voltages; id and iq are stator currents; Ld and Lq are stator inductances; ψf is magnetic flux; R is stator resistance; p is pole-pair numbers in the PMSM; J is inertia moment; B is friction modulus; ωr is angular velocity of the rotor; Te is electromagnetic torque; TL is load torque.To simplify the mathematical expression, we define(2)X1=id,iqT;X2=F=fd,fqT,F˙=ϑ=ϑ1,ϑ2T,U=ud,uqT,H=Lpiqωr,Lpidωr−ψfpωrT,Y=X1,where maxϑ1,ϑ2T<l. Subsequently, by substituting equation (2) into equation (1), we obtain(3)X˙1=1L−RX1+H+U+X2,X˙2=ϑ.


### 2.1. Current Loop FCSOC Controller of PMSM

The overall control diagram of the PMSM servo system is shown in Figure1. It can be seen that the SESO and FCSOC are applied in the current control loop. The SESO is an ESO characterized with high-precision observation in changing disturbance and states with the super-twisting algorithm, which can improve the anti-interference ability of IFCSOC. When disturbance exists, the adaptive disturbance compensation FCSOC performs better on dynamic response.Figure 1
Diagram of a PMSM control system.


### 2.2. Design of the SESO

The SESO is a general ESO combined with the super-twisting algorithm; the observation of ESO is(4)Z1=id∗,iq∗T,Z2=fd∗,fq∗T,I¯=X1−Z1,F¯=X2−Z2,σ1=I¯+RLI¯+qI¯λsgnI¯,where Z1 is the observation of X1, Z2 is the estimation of disturbanceF, I¯ and F¯ are estimations of transient errors, σ1 is a variable in sliding-mode control, and λ is a constant 1/2<λ<1.To simplify the mathematical expression, the following notions are claimed:(5)sgnwρ=w1ρ1sgnw1,…,wnρnsgnwnT,μsgnwρ=μ1ρ1sgnw1,…,μnρnsgnwnT.The SESO is constructed as(6)Z˙1=1L−RX1+H+U+Z2−δ1Θ1I¯,Z˙2=κI¯I¯⋅−δ2Θ2I¯,where(7)Θ1I¯=I¯1/2signI¯+I¯,Θ2I¯=12sgnI¯+I¯+32I¯1/2sgnI¯,κI¯=λ+1qI¯λsignI¯.Note thatδ1 and δ2 are two parameters to be adjusted. According to (3) and (6)fd6, the corresponding errors in observation of the SESO (6) can be expressed as follows:(8)I¯⋅=F¯−δ1Θ1I¯,F¯⋅=κI¯I¯⋅−δ2Θ2I¯.Assume thatI¯≠0,0T in the initial state. Set V0 as Lyapunov function:(9)V0ϑt=ϑtTPϑt,where ϑtT=Θ1I¯,F¯, in which ϑ˙t is expressed as follows:(10)ϑ˙t=Θ1′I¯−δ1Θ1I¯+F¯−δ2Θ2I¯+κI¯I¯⋅Θ1′I¯=Θ1′Aϑt+Bεt,where εt=κI¯I¯⋅/Θ1′I¯, A=−δ11−δ20, Θ1′I¯=I¯−1/2/2+1, and B=01.Assumption 1.
(see [22]). The disturbance εt and ϑt∈R2 satisfy the following inequality:(11)Λϑt,εt=ϑtεtHSTS−Rϑtεt≥0,where H≥0, R≥0, S=cm, and c and m are two constants.
The convergence time of SESO in equation (8) can be determined in finite time; for the details, refer to the following theorem.Theorem 1.
(see [23]). If the following inequality holds,(12)ATP+PA+τP+HPB+STBTP+S−R≤0,the allowable convergence time can be represented as(13)Ts=2τln2λminPV01/2ϑ0+1,where τ and R are positive constants and P is a positive and symmetric matrix. In this condition, the observation of SESO in (8) can realize finite-time stability; ϑ0 is the initial condition.


### 2.3. Disturbance Compensation FCSOC for PMSM

The disturbance estimation is performed in the previous section. FCSOC is adopted to track the reference currenti^d and i^q. The current errors can be expressed as(14)e˙de˙q=i^d−idi^q−iq.The differentiation of equation (14) is(15)e˙de˙q=di^ddt−1L−Rid+Lpiqwr+ud−fddi^qdt−1L−Rid+Lpidwr−ψfpwr+uq−fq.To compensate the disturbances (which has been estimated before), the control law is(16)ud∗uq∗=Ldi^ddt+Rid−Lpiqwr+k1edηsgned−Lf^dLdi^qdt+Riq−Lpidwr+ψfpwr+k2eqηsgneq−Lf^q.Substituted by equation (16), equation (15) is transformed as(17)e˙de˙q=−f˜d−k1edηsgnedL−f˜q−k2eqηsgneqL,where f˜d and f˜q are estimation errors:(18)f˜df˜q=fd−f^dfq−f^q.Set Lyapunov functionV1 as(19)V1=ed2+eq2+12γ1f˜d2+12γ2f˜q2,where γ2 and γ3 are positive parameters.The derivative of equation (19) is(20)V˙1=ede˙d+eqe˙q+1γ1f˜df˜⋅d+1γ2f˜qf˜⋅q=ed−f˜d−k1edηsgnedL+eq−f˜q−k2eqηsgneqL+1γ1f˜df˜⋅d+1γ2f˜qf˜⋅q=−k1Ledη+1−k2Leqη+1+f˜d1γ1f˜⋅d−ed+f˜q1γ2f˜⋅q−eq.To compensate disturbances, control laws are given as(21)f˜⋅d=γ1ed,f˜⋅q=γ2eq.By using of the adaptive law in equation (21), the estimation of f˜d and f˜q can be more accurate.


## 2.1. Current Loop FCSOC Controller of PMSM

The overall control diagram of the PMSM servo system is shown in Figure1. It can be seen that the SESO and FCSOC are applied in the current control loop. The SESO is an ESO characterized with high-precision observation in changing disturbance and states with the super-twisting algorithm, which can improve the anti-interference ability of IFCSOC. When disturbance exists, the adaptive disturbance compensation FCSOC performs better on dynamic response.Figure 1
Diagram of a PMSM control system.


## 2.2. Design of the SESO

The SESO is a general ESO combined with the super-twisting algorithm; the observation of ESO is(4)Z1=id∗,iq∗T,Z2=fd∗,fq∗T,I¯=X1−Z1,F¯=X2−Z2,σ1=I¯+RLI¯+qI¯λsgnI¯,where Z1 is the observation of X1, Z2 is the estimation of disturbanceF, I¯ and F¯ are estimations of transient errors, σ1 is a variable in sliding-mode control, and λ is a constant 1/2<λ<1.To simplify the mathematical expression, the following notions are claimed:(5)sgnwρ=w1ρ1sgnw1,…,wnρnsgnwnT,μsgnwρ=μ1ρ1sgnw1,…,μnρnsgnwnT.The SESO is constructed as(6)Z˙1=1L−RX1+H+U+Z2−δ1Θ1I¯,Z˙2=κI¯I¯⋅−δ2Θ2I¯,where(7)Θ1I¯=I¯1/2signI¯+I¯,Θ2I¯=12sgnI¯+I¯+32I¯1/2sgnI¯,κI¯=λ+1qI¯λsignI¯.Note thatδ1 and δ2 are two parameters to be adjusted. According to (3) and (6)fd6, the corresponding errors in observation of the SESO (6) can be expressed as follows:(8)I¯⋅=F¯−δ1Θ1I¯,F¯⋅=κI¯I¯⋅−δ2Θ2I¯.Assume thatI¯≠0,0T in the initial state. Set V0 as Lyapunov function:(9)V0ϑt=ϑtTPϑt,where ϑtT=Θ1I¯,F¯, in which ϑ˙t is expressed as follows:(10)ϑ˙t=Θ1′I¯−δ1Θ1I¯+F¯−δ2Θ2I¯+κI¯I¯⋅Θ1′I¯=Θ1′Aϑt+Bεt,where εt=κI¯I¯⋅/Θ1′I¯, A=−δ11−δ20, Θ1′I¯=I¯−1/2/2+1, and B=01.Assumption 1.
(see [22]). The disturbance εt and ϑt∈R2 satisfy the following inequality:(11)Λϑt,εt=ϑtεtHSTS−Rϑtεt≥0,where H≥0, R≥0, S=cm, and c and m are two constants.
The convergence time of SESO in equation (8) can be determined in finite time; for the details, refer to the following theorem.Theorem 1.
(see [23]). If the following inequality holds,(12)ATP+PA+τP+HPB+STBTP+S−R≤0,the allowable convergence time can be represented as(13)Ts=2τln2λminPV01/2ϑ0+1,where τ and R are positive constants and P is a positive and symmetric matrix. In this condition, the observation of SESO in (8) can realize finite-time stability; ϑ0 is the initial condition.


## 2.3. Disturbance Compensation FCSOC for PMSM

The disturbance estimation is performed in the previous section. FCSOC is adopted to track the reference currenti^d and i^q. The current errors can be expressed as(14)e˙de˙q=i^d−idi^q−iq.The differentiation of equation (14) is(15)e˙de˙q=di^ddt−1L−Rid+Lpiqwr+ud−fddi^qdt−1L−Rid+Lpidwr−ψfpwr+uq−fq.To compensate the disturbances (which has been estimated before), the control law is(16)ud∗uq∗=Ldi^ddt+Rid−Lpiqwr+k1edηsgned−Lf^dLdi^qdt+Riq−Lpidwr+ψfpwr+k2eqηsgneq−Lf^q.Substituted by equation (16), equation (15) is transformed as(17)e˙de˙q=−f˜d−k1edηsgnedL−f˜q−k2eqηsgneqL,where f˜d and f˜q are estimation errors:(18)f˜df˜q=fd−f^dfq−f^q.Set Lyapunov functionV1 as(19)V1=ed2+eq2+12γ1f˜d2+12γ2f˜q2,where γ2 and γ3 are positive parameters.The derivative of equation (19) is(20)V˙1=ede˙d+eqe˙q+1γ1f˜df˜⋅d+1γ2f˜qf˜⋅q=ed−f˜d−k1edηsgnedL+eq−f˜q−k2eqηsgneqL+1γ1f˜df˜⋅d+1γ2f˜qf˜⋅q=−k1Ledη+1−k2Leqη+1+f˜d1γ1f˜⋅d−ed+f˜q1γ2f˜⋅q−eq.To compensate disturbances, control laws are given as(21)f˜⋅d=γ1ed,f˜⋅q=γ2eq.By using of the adaptive law in equation (21), the estimation of f˜d and f˜q can be more accurate.


## 3. The SADRC Controller in Position Loop

### 3.1. The Structure of SADRC Controller

In the overall IFCSOC system, a SADRC controller is adopted in position control. The details are shown in Figure2. Generally, considering id∗=0 in a synchronously rotating coordinate system, the position loop is expressed as(22)d2θitdt2=−BJθ˙it+1.5npψfJiq∗−TLJ,where iq∗ and iq are the reference current and real current of the stator in the q coordinate axis, respectively; θit is the rotor position; TL is the load torque; ψf is the rotor flux; np is the number of pole pairs; J is the inertia of the rotor; B is the friction parameter.Figure 2
Structure of the SADRC-FCSOC controller.Set new state variables asx1=θit, x2=dθit/dt=ω,u=iq, b=1.5npψf/J, f1x1,x2,t=−TL/J, and f0x1,x2=−B/Jθ˙it. The mathematical model of position loop can be expressed as(23)x˙1=x2=dθitdt,x˙2=d2θitdt2=f0x1,x2+f1x1,x2,t+bu,g=x1=dθitdt,where x1 is the tracking signal of θi∗; x2is the differentiation of x1; f0x1,x2 is the known disturbance; f1x1,x2,t is the unknown external disturbance; ωt=f0x1,x2+f1x1,x2,t is the total disturbance.In the SADRC controller, the nonlinear tracking differentiator (NTD) is used to track state variablesx1 and x2. z1, z2 , and z3 are the system state values and disturbance estimated by using the ESO. e1=x1−z1 represents the tracking error. The integral and differentiation of tracking errors are e0=∫e1dt and e2=x2−z2. From Figure 2, we can obtain reference current in the following form:(24)iq∗=β0fale0,α5,δ4+β1fale1,α1,δ1+β2fale2,α2,δ2−z3+f0z1,z2b,where β0, β1, and β2 are the e0, e1, and e2 gains of NLSEF, respectively. The nonlinear function falek,α,δ can be defined as(25)falek,α,δ=ekαsgnek,ek>δ,ekδ1−α,ek≤δ.


### 3.2. Taylor’s Formula Compensation for the Nonlinear Tracking Differentiator (NTD)

The nonlinear tracking differentiator (NTD) in ADRC is used to track reference signal and derivative of the reference signal. The NTD process can be given as(26)x1k+1=x1k+hx2k,x2k+1=x2k+hfstx1k−gdk,x2k,r,h0,where gdk is the desired trajectory; h0 is the filter coefficient; r is the speed coefficient; h is the sampling step; x1k and x2k are the tracking signal of gdk andg˙dk.Generally, tracking reference can be improved by raisingrand h0. The definition of nonlinear function fst⋅ is(27)fstx1k,x2k,r,h=−rad,a≤d,−rsgna,a>d,a=x2k+a0−d2sgnyk,yk>d0,x2k+ykh,yk≤d0,d=rh,d0=dh,a0=d2+8ryk,gk=x1k+hx2k.According to Taylor’s formula, the value ofgdk at k+Δk can be predicted as(28)gdk+Δk=gdk+gd′k1!Δk+gd′′k2!Δk2+⋯+gdnkn!Δkn+gdn+1ξn+1!Δkn+1.In Figure3, gdk is the true signal and x1k is the tracking signal, we can see that there is a phase delay between the tracking signal and the true signal. Phase delay happens because the tracking speed is not appropriate all the time. From Figure 3, the true signal gdk1 equals the tracking signal x1k2. Similarly, the true signal at time kn equals the tracking signal at kn+1, the delay time Δk=kn+1−kn. The value of tracking signal may be smaller or larger than that of true signal. For example, x1k1 is larger than gdk1, while x1kn is smaller than gdkn. Therefore, Taylor’s formula is used in the DNTD to compensate the phase delay; the method can be expressed as(29)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.Figure 3
The schematic diagram of phase delay.If we obtain the delay timeΔk=kn+1−kn=rCh, we can make use of Taylor’s formula to compensate retardation for accuracy tracking. In this paper, we set the compensate step as rC. Then, equation (29) can be rewritten as(30)g˜dkn+1=x1kn+1+x2kn+11!kn+1−kn+x3kn+12!kn+1−kn2+⋯+xnkn+1n!kn+1−knn.Taylor’s approach is expected to compensate phase delay based on the aforementioned theory; the method in [24] is given as(31)gdkn+1=x1kn+1+rCΔk⋅x2kn+1.Subsequently, equation (31) can be expressed as(32)gk+1=x1k+1+rCΔk⋅x2k+1,x2k+1=x2k+hfstx1k−gdk,x2k,r,h0.SetrC as the time step in compensation; M is the maximum value of fn+1ξ in kn,kn+1, and the Lagrange remainder Rn of the proposed Taylor’s approach should meet(33)Rn≤Mn+1!x−x0n+1.


### 3.3. Nonlinear Extended State Observer (NESO)

NESO can estimate disturbances quickly and accurately. Therefore, this paper combines the disturbance observer (DOB) with the ESO to derive the NESO. Letx1=θit and x2=dθit/dt=ω, then(34)x˙1=x2=dθitdt,x˙2=d2θitdt2=f0x1,x2+f1x1,x2,t+bu=fx1,x2,t+biq.Choose a new state as(35)x2=fx1,x2,t,x˙2=Lt,with Lt=f˙x1,x2,t.The NESO in ADRC is designed to estimate all the state variables and disturbances.(36)ek=z1k−yk,z1k+1=z1k−hz2k−β01ek,z2k+1=z3k−β02falek,α1,δ1+giq,z˙3k+1=−lz3k−llwr+fx+giq,f^=z3k+lwr,where fx=−B/Jwr, gx=3npφf/2J, and l is the gain matrix of NESO.


### 3.4. Sliding-Mode Nonlinear State Error Feedback (SM-NLSEF)

To improve the response speed and simplify the parameter tuning, we select functionge2k instead of nonlinear operator function falek,α,δ:(37)e1k+1=x1k+1−z1k+1,e2k+1=x2k+1−z2k+1,e˙2=ge2,iq∗=ge2−z3.Design a sliding-mode surfacest=ce2 and sliding function ge2=−ksgns. According to Lyapunov stability theory, lims⟶0ss˙<0 should be satisfied.(38)st⋅s˙t=st−ksgnst.Sincek>0 is a tunable parameter, the system is always stable. To suppress the chattering in the system and enhance the control continuity, the anti-chattering factor Gs is introduced to replace sgns:(39)Gs=ss+v,where v is the chattering factor, with v>0. The SM-NLSEF is represented as(40)e1k+1=x1k+1−z1k+1,e2k+1=x2k+1−z2k+1,iq∗=−kGs−z3.


### 3.5. Simulation and Analysis

To further explore the effectiveness of SADRC-FCSOC controller, we carry out the simulation studies of step response and sinusoidal signal tracking. The parameters of PMSM are listed in Table1. The NTD parameters are h=0.01 and rC=1000. The NESO parameters are β01=1 and β02=15.8. The SESO parameters are set as δ1=δ2=0.01. The rational parameters of FCSOC are set as k1=k2=100, η=0.5, and γ1=γ2=3.5.Table 1
Parameters in the 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 ratioi1039Stator inductanceLd=Lq21.24mHPole pairsnp3Motor magnetic fluxψf0.65WbFigure4 shows the step response of IFCSOC and FCSOC with a 500 nm load added at 6.0 s. The simulation data are recorded in Table 2.Figure 4
Step simulation with load step disturbance.Table 2
Step simulation data.
Control strategyAdjustment time (s)Maximum disturbance (°)Recovery time (s)Steady-state error (°)IFCSOC1.230.340.640.032FCSOC1.670.780.850.056From Figure4, it can be seen that IFCSOC exhibits a faster response than FCSOC. When a disturbance is added at 6.0 s, IFCSOC has a smaller position offset response. Furthermore, IFCSOC has better robust performance over FCSOC. The maximum disturbance value of IFCSOC is 0.34° smaller than FCSOC. Also, IFCSOC has better steady-state accuracy. The steady-state error of IFCSOC is 3.2%; meanwhile, the steady-state error of FCSOC is 5.6%, which is 1.75 times higher than that of IFCSOC.Figure5 shows the step response of IFCSOC and FCSOC with rapidly changing disturbances, where the maximum steady-state errors of the IFCSOC controller and FCSOC controller are 0.049° and 0.082°, respectively. It is indicated that the IFCSOC controller has stronger anti-interference ability than the FCSOC.Figure 5
Step simulation with load rapidly changing disturbance.To measure the dynamic tracking performance of IFCSOC, a sinusoidal signal tracking simulation is conducted. As shown in Figure6, IFCSOC achieves a higher accuracy compared with FCSOC in tracking the sinusoidal signal. The phase lags have been reduced when adopting the IFCSOC method, which means Taylor’s formula to compensate phase delay is effective in raising tracking efficiency.Figure 6
Tracking of the sinusoidal signal.Figure7 shows the tracking errors of sinusoidal signal, from which it can be seen that IFCSOC has a better tracking performance compared with FCSOC. The maximum tracking error of IFCSOC is 0.038°, while the FCSOC can reach 0.117°.Figure 7
Tracking errors of the sinusoidal signal.


## 3.1. The Structure of SADRC Controller

In the overall IFCSOC system, a SADRC controller is adopted in position control. The details are shown in Figure2. Generally, considering id∗=0 in a synchronously rotating coordinate system, the position loop is expressed as(22)d2θitdt2=−BJθ˙it+1.5npψfJiq∗−TLJ,where iq∗ and iq are the reference current and real current of the stator in the q coordinate axis, respectively; θit is the rotor position; TL is the load torque; ψf is the rotor flux; np is the number of pole pairs; J is the inertia of the rotor; B is the friction parameter.Figure 2
Structure of the SADRC-FCSOC controller.Set new state variables asx1=θit, x2=dθit/dt=ω,u=iq, b=1.5npψf/J, f1x1,x2,t=−TL/J, and f0x1,x2=−B/Jθ˙it. The mathematical model of position loop can be expressed as(23)x˙1=x2=dθitdt,x˙2=d2θitdt2=f0x1,x2+f1x1,x2,t+bu,g=x1=dθitdt,where x1 is the tracking signal of θi∗; x2is the differentiation of x1; f0x1,x2 is the known disturbance; f1x1,x2,t is the unknown external disturbance; ωt=f0x1,x2+f1x1,x2,t is the total disturbance.In the SADRC controller, the nonlinear tracking differentiator (NTD) is used to track state variablesx1 and x2. z1, z2 , and z3 are the system state values and disturbance estimated by using the ESO. e1=x1−z1 represents the tracking error. The integral and differentiation of tracking errors are e0=∫e1dt and e2=x2−z2. From Figure 2, we can obtain reference current in the following form:(24)iq∗=β0fale0,α5,δ4+β1fale1,α1,δ1+β2fale2,α2,δ2−z3+f0z1,z2b,where β0, β1, and β2 are the e0, e1, and e2 gains of NLSEF, respectively. The nonlinear function falek,α,δ can be defined as(25)falek,α,δ=ekαsgnek,ek>δ,ekδ1−α,ek≤δ.


## 3.2. Taylor’s Formula Compensation for the Nonlinear Tracking Differentiator (NTD)

The nonlinear tracking differentiator (NTD) in ADRC is used to track reference signal and derivative of the reference signal. The NTD process can be given as(26)x1k+1=x1k+hx2k,x2k+1=x2k+hfstx1k−gdk,x2k,r,h0,where gdk is the desired trajectory; h0 is the filter coefficient; r is the speed coefficient; h is the sampling step; x1k and x2k are the tracking signal of gdk andg˙dk.Generally, tracking reference can be improved by raisingrand h0. The definition of nonlinear function fst⋅ is(27)fstx1k,x2k,r,h=−rad,a≤d,−rsgna,a>d,a=x2k+a0−d2sgnyk,yk>d0,x2k+ykh,yk≤d0,d=rh,d0=dh,a0=d2+8ryk,gk=x1k+hx2k.According to Taylor’s formula, the value ofgdk at k+Δk can be predicted as(28)gdk+Δk=gdk+gd′k1!Δk+gd′′k2!Δk2+⋯+gdnkn!Δkn+gdn+1ξn+1!Δkn+1.In Figure3, gdk is the true signal and x1k is the tracking signal, we can see that there is a phase delay between the tracking signal and the true signal. Phase delay happens because the tracking speed is not appropriate all the time. From Figure 3, the true signal gdk1 equals the tracking signal x1k2. Similarly, the true signal at time kn equals the tracking signal at kn+1, the delay time Δk=kn+1−kn. The value of tracking signal may be smaller or larger than that of true signal. For example, x1k1 is larger than gdk1, while x1kn is smaller than gdkn. Therefore, Taylor’s formula is used in the DNTD to compensate the phase delay; the method can be expressed as(29)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.Figure 3
The schematic diagram of phase delay.If we obtain the delay timeΔk=kn+1−kn=rCh, we can make use of Taylor’s formula to compensate retardation for accuracy tracking. In this paper, we set the compensate step as rC. Then, equation (29) can be rewritten as(30)g˜dkn+1=x1kn+1+x2kn+11!kn+1−kn+x3kn+12!kn+1−kn2+⋯+xnkn+1n!kn+1−knn.Taylor’s approach is expected to compensate phase delay based on the aforementioned theory; the method in [24] is given as(31)gdkn+1=x1kn+1+rCΔk⋅x2kn+1.Subsequently, equation (31) can be expressed as(32)gk+1=x1k+1+rCΔk⋅x2k+1,x2k+1=x2k+hfstx1k−gdk,x2k,r,h0.SetrC as the time step in compensation; M is the maximum value of fn+1ξ in kn,kn+1, and the Lagrange remainder Rn of the proposed Taylor’s approach should meet(33)Rn≤Mn+1!x−x0n+1.


## 3.3. Nonlinear Extended State Observer (NESO)

NESO can estimate disturbances quickly and accurately. Therefore, this paper combines the disturbance observer (DOB) with the ESO to derive the NESO. Letx1=θit and x2=dθit/dt=ω, then(34)x˙1=x2=dθitdt,x˙2=d2θitdt2=f0x1,x2+f1x1,x2,t+bu=fx1,x2,t+biq.Choose a new state as(35)x2=fx1,x2,t,x˙2=Lt,with Lt=f˙x1,x2,t.The NESO in ADRC is designed to estimate all the state variables and disturbances.(36)ek=z1k−yk,z1k+1=z1k−hz2k−β01ek,z2k+1=z3k−β02falek,α1,δ1+giq,z˙3k+1=−lz3k−llwr+fx+giq,f^=z3k+lwr,where fx=−B/Jwr, gx=3npφf/2J, and l is the gain matrix of NESO.


## 3.4. Sliding-Mode Nonlinear State Error Feedback (SM-NLSEF)

To improve the response speed and simplify the parameter tuning, we select functionge2k instead of nonlinear operator function falek,α,δ:(37)e1k+1=x1k+1−z1k+1,e2k+1=x2k+1−z2k+1,e˙2=ge2,iq∗=ge2−z3.Design a sliding-mode surfacest=ce2 and sliding function ge2=−ksgns. According to Lyapunov stability theory, lims⟶0ss˙<0 should be satisfied.(38)st⋅s˙t=st−ksgnst.Sincek>0 is a tunable parameter, the system is always stable. To suppress the chattering in the system and enhance the control continuity, the anti-chattering factor Gs is introduced to replace sgns:(39)Gs=ss+v,where v is the chattering factor, with v>0. The SM-NLSEF is represented as(40)e1k+1=x1k+1−z1k+1,e2k+1=x2k+1−z2k+1,iq∗=−kGs−z3.


## 3.5. Simulation and Analysis

To further explore the effectiveness of SADRC-FCSOC controller, we carry out the simulation studies of step response and sinusoidal signal tracking. The parameters of PMSM are listed in Table1. The NTD parameters are h=0.01 and rC=1000. The NESO parameters are β01=1 and β02=15.8. The SESO parameters are set as δ1=δ2=0.01. The rational parameters of FCSOC are set as k1=k2=100, η=0.5, and γ1=γ2=3.5.Table 1
Parameters in the 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 ratioi1039Stator inductanceLd=Lq21.24mHPole pairsnp3Motor magnetic fluxψf0.65WbFigure4 shows the step response of IFCSOC and FCSOC with a 500 nm load added at 6.0 s. The simulation data are recorded in Table 2.Figure 4
Step simulation with load step disturbance.Table 2
Step simulation data.
Control strategyAdjustment time (s)Maximum disturbance (°)Recovery time (s)Steady-state error (°)IFCSOC1.230.340.640.032FCSOC1.670.780.850.056From Figure4, it can be seen that IFCSOC exhibits a faster response than FCSOC. When a disturbance is added at 6.0 s, IFCSOC has a smaller position offset response. Furthermore, IFCSOC has better robust performance over FCSOC. The maximum disturbance value of IFCSOC is 0.34° smaller than FCSOC. Also, IFCSOC has better steady-state accuracy. The steady-state error of IFCSOC is 3.2%; meanwhile, the steady-state error of FCSOC is 5.6%, which is 1.75 times higher than that of IFCSOC.Figure5 shows the step response of IFCSOC and FCSOC with rapidly changing disturbances, where the maximum steady-state errors of the IFCSOC controller and FCSOC controller are 0.049° and 0.082°, respectively. It is indicated that the IFCSOC controller has stronger anti-interference ability than the FCSOC.Figure 5
Step simulation with load rapidly changing disturbance.To measure the dynamic tracking performance of IFCSOC, a sinusoidal signal tracking simulation is conducted. As shown in Figure6, IFCSOC achieves a higher accuracy compared with FCSOC in tracking the sinusoidal signal. The phase lags have been reduced when adopting the IFCSOC method, which means Taylor’s formula to compensate phase delay is effective in raising tracking efficiency.Figure 6
Tracking of the sinusoidal signal.Figure7 shows the tracking errors of sinusoidal signal, from which it can be seen that IFCSOC has a better tracking performance compared with FCSOC. The maximum tracking error of IFCSOC is 0.038°, while the FCSOC can reach 0.117°.Figure 7
Tracking errors of the sinusoidal signal.


## 4. Semiphysical Simulation Experiment

To evaluate the performance of IFCSOC controller, a semiphysical simulation is conducted in our laboratory for position control of the PMSM. The overall view of semiphysical workbench is illustrated in Figure8 and listed in Table 3.Figure 8
Semiphysical workbench structure.
(a)(b)Table 3
Components of the semiphysical simulation workbench.
AbbreviationFull namePAPower amplifierPRGPrecision reduction gearboxLFLoading fixtureRIPRotational inertia plateMPBMagnetic power brakeAMActuating motorThe workbench is composed of the following parts: the control computer, sensor system, PA, PRG, LF, AM, RIP, and the support. The RIP and MPB are used to simulate disturbance factors in the system, such as frictional moment, load torque, and rotational inertia. By controlling the RIP, the rotational inertia can be simulated. In the same way, the MPB output torque can simulate rapidly changing torque and frictional moment.To verify the dynamic tracking accuracy of the IFCSOC method applied in the servo system, an experiment is conducted on the semiphysical workbench for tracking a sinusoidal signal with a period of 3.768 s and an amplitude of30°. The tracking curves of FCSOC and IFCSOC are shown in Figure 9. It can be seen that the tracking curve of IFCSOC is more accurate than that of FCSOC. Also, IFCSOC has a smaller phase delay.Figure 9
Comparison diagram of sinusoidal tracking curves.Figure10 compares the tracking errors obtained from preliminary experiments. It implies that, in the initial tracking period, the maximum tracking error of FCSOC is 0.53° and IFCSOC is 0.58°. Entering the stabilization stage, the maximum tracking error of FCSOC is 0.203° and IFCSOC is 0.073°. Also, it can be seen that the chattering amplitude of IFCSOC is relatively smaller than that of FCSOC, showing a better tracking performance.Figure 10
Diagram of sinusoidal tracking error curves.Figure11 shows the angular velocity and acceleration curves of the IFCSOC controller. The chattering of velocity and acceleration is reasonable. The proposed control method can be applied to the shipborne rocket launcher system.Figure 11
Velocity and acceleration curves.


## 5. Conclusion

The proposed IFCSOC method can deal with the nonlinearities and rapidly changing disturbances caused by complex sea conditions of the rocket launcher system. Three comparative simulations are carried out, showing that the new method can improve robustness and tracking accuracy of the system. Finally, the semiphysical experiment validates the effectiveness of IFCSOC strategy.



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*Source: 3651298-2020-08-07.xml*