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Research article Special Issues

RISE-based adaptive control of electro-hydraulic servo system with uncertain compensation


  • Electro-hydraulic servo system (EHSS) plays an important role in many industrial and military applications. However, its high-performance tracking control is still a challenging mission due to its nonlinear system dynamics and model uncertainties. In this paper, a novel adaptive robust integral method of the sign of the error (ARISE) with extended state observer (ESO) is proposed. Firstly, the nonlinear mathematical model of typical EHSS with modeling uncurtains and uncertain nonlinear is established. Then, ESO is used to estimate the state and lumped disturbance, of which the unknown parameter estimations can be updated by the novel adaptive law. Results shows that the novel controller achieves better tracking performance in maximum tracking error, average tracking error and standard deviation of the tracking error.

    Citation: Xiaohan Yang, Yinghao Cui, Zhanhang Yuan, Jie Hang. RISE-based adaptive control of electro-hydraulic servo system with uncertain compensation[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 9288-9304. doi: 10.3934/mbe.2023407

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  • Electro-hydraulic servo system (EHSS) plays an important role in many industrial and military applications. However, its high-performance tracking control is still a challenging mission due to its nonlinear system dynamics and model uncertainties. In this paper, a novel adaptive robust integral method of the sign of the error (ARISE) with extended state observer (ESO) is proposed. Firstly, the nonlinear mathematical model of typical EHSS with modeling uncurtains and uncertain nonlinear is established. Then, ESO is used to estimate the state and lumped disturbance, of which the unknown parameter estimations can be updated by the novel adaptive law. Results shows that the novel controller achieves better tracking performance in maximum tracking error, average tracking error and standard deviation of the tracking error.



    Ap Ram area
    Ct Internal leakage coefficient
    P1, P2, Oil pressures inside the chambers of the actuator, respectively
    PL Load pressure
    Pr Return pressure
    Ps Supply pressure
    Q1, Q2 Oil flow in both chambers of the actuator
    V01, V02 Initial volume of oil inlet and outlet chamber, respectively
    V1, V2 Control volume of oil inlet and outlet chamber, respectively
    fv Damping coefficient
    g1, g2 Pressure difference of inlet and outlet
    k Spring stiffness
    k1, k2, k3, kr Positive gains
    kt Total gain of the servo valve
    mp Mass of load
    q1, q2 Model errors because of internal leakage
    u Control voltage
    xp Position of load
    β Gain
    βe1, βe2 Oil effective bulk modulus of the actuator, respectively
    ω Bandwidth of the extended state observer
    ARISE Adaptive robust integral of the sign of the error
    ESO Extended state observer
    RISE Robust integral of the sign of the error

    Electric hydraulic servo system (EHSS) is widely used in industry applications due to its high power ratio and fast response [1,2,3,4,5,6]. With the rapid development of technology, more severe control requirements are put forward for EHSS, and high-performance control becomes eagerly needed to address its behavioral nonlinearity and uncertainties, such as flow nonlinearity, pressure dynamic nonlinearity, uncertain parameters, unmolded nonlinearity, and external disturbances [7,8,9,10].

    Nowadays, many researchers focus on advanced control strategies such as robust adaptive control [11,12], sliding mode control [13,14], back-stepping control [15,16], robust integral of the sign of the error (RISE) [17,18] for EHSS. Yue and Yao [17] proposed an adaptive robust integral of the sign of the error control (ARISE), which can adjust the robust gain online through adaptive method to solve the potential high gain feedback of symbolic function. Yao et al. [19] designed a novel ARSE to address noise pollution in the acquisition of acceleration signal, which can compensate the error of friction model and other bounded disturbances [20]. The above literature shows that RISE/ARISE control greatly improves the trajectory tracking accuracy of EHSS. However, those controllers regard the parameter adaptive error, unmolded error, and external disturbance as lumped disturbance, which severely limits the control accuracy. Control strategy based on disturbance observer can compensate the influence of disturbance and uncurtains effectively, which has been used in the field of control theory and engineering [21,22,23,24,25,26]. Especially, the extended state observer (ESO), which is the core of ADRC and has been widely used in disturbance estimation and suppression [27,28,29,30,31,32].

    In this paper, the sign function is replaced with a modified arctangent function to smooth the nonlinearity of sign function. Then, the state and external disturbance can be estimated by ESO, respectively. Finally, the residual observation error is compensated to further enhance the tracking accuracy by ARISE. The Lyapunov theory proves the EHSS can achieve asymptotic s. Simulation results show the proposed controller has a better performance in maxi-mum tracking error, average tracking error and standard deviation of the tracking error.

    The typical working principle of EHSS is shown in Figure 1. The double rod symmetrical hydraulic cylinder is controlled by servo valve to drive the load.

    Figure 1.  Model of valve controlled symmetrical cylinder system.

    In this paper, the force balance equation of EHSS can be given as

    mp¨xp=PLApkxpfv˙xp+f(xp,˙xp,t) (1)

    where mp and xp represent mass and displacement of load respectively; PL is the pressure difference; Ap is the effective area of the piston; k is the spring stiffness; fv is the combined coefficient; f(xp,˙xp,t) indicates the lumped uncertain; Pressure dynamics of the two chambers are given by [15].

    {˙P1=βe1/βe1V1V1(Q1Ap˙xpCtPL+q1(t))˙P2=βe2/βe2V2V2(Ap˙xp+CtPLQ2q2(t)) (2)

    where

    {Q1=ktg1(P1,xv)uQ2=ktg2(P2,xv)u (3)
    {g1=(PsPt+(Ps2P1+Pt)s(u))/2g2=(PsPt(Ps2P2+Pt)s(u))/2 (4)

    The pressure difference dynamics between the two chambers is expressed as follows

    ˙PL=βekt(g1/g1V1+g2/g2V2V2V1+g2/g2V2V2)uβeAp(1/1V1+1/1V2V2V1+1/1V2V2)˙xpβeCt(1/1V1+1/1V2V2V1+1/1V2V2)PL+βe(q1/q1V1+q2/q2V2V2V1+q2/q2V2V2) (5)

    where βe1, βe2 are the effective elastic modulus in two chamber and βe1 = βe2 = βe; V1 = V01 + Ap; V2 = V02 - Ap·xp represents the control volume of return chamber; V01 and V02 are the initial volumes of the two chambers respectively; Ct is the internal leakage coefficient of the cylinder; Q1 (t) and Q2 (t) are the oil flowrate of the two chamber of the cylinder respectively; q1(t) and q2(t) are model errors because of internal leakage in the two chambers; g1 is the pressure difference at the oil inlet and g2 is the pressure difference at the oil outlet; Ps is the supplied pressure; Pr is the return pressure, kt is the total gain of the servo valve; u is the control voltage and s(u) is expressed as

    s(u)={1u>00u=01u<0 (6)

    To make it smooth and differentiable, the sign function s(u) is replaced by Eq (7).

    farctan(u)=2arctan(Ku)/π (7)

    Thus, g1, g2 in Eq (4) can be rewritten as Eq (8).

    {g1=(PsPt+(Ps2P1+Pt)farctan(u))/2g2=(PsPt(Ps2P2+Pt)farctan(u))/2 (8)

    Substituting Eqs (2)-(5) into Eq (1), and thus

    mpxp=Apβukl(g1/V1+g2/V2)uA2pβu(1/V1+1/V2)ˉxpApβeCt(1/V1+1/V2)PL+Apβu(q1/V1+q2/V2)k˙xpfv¨xp+˙f (9)

    Rewritten Eq (9), and thus,

    mpxp=Apβukt(g1/V1+g2/V2)uA2pβe(1/V1+1/V2)ˉxpApβeCt(1/V1+1/V2)FL+Apβe(q1/V1+q2/V2)k˙xpfv¨xp+˙f+f (10)

    where f is the approximation error caused by using the continuously differentiable function farctan(u).

    According to Eq (1)

    PL=mpAp¨xp+kApxp+fvAp˙xpfAp (11)

    Substituting Eq (11) into Eq (10), thus

    mpxp=Apβekt(g1/V1+g2/V2)uβeCt(1/V1+1/V2)kxp(A2pβe(1/V1+1/V2)+fvβeCt(1/V1+1/V2)+k)˙xp(mpβeCt(1/V1+1/V2)+fv)¨xp+βeCt(1/V1+1/V2)f+˙f+f+Apβe(q1/V1+q2/V2) (12)

    Define state variables as x=[x1,x2,x3]T=[xp,˙xp,¨xp]T and output variables as y=x1=xp, so the state space model of EHSS can be expressed as

    {˙x1=x2˙x2=x3˙x3=θ1uθ2x1θ3x2θ4x3+Δ (13)

    where θ1=Apβekt(g1V1+g1V2)/mp,θ2=βeCt(1V1+1V2)k/mp, θ3=(A2pβe(1V1+1V2)+fvβeCt(1V1+1V2)+k)/mp, θ4=(mpβeCt(1V1+1V2)+fv)/mp,Δ=(βeCt(1V1+1V2)f+˙f+f+Apβe(q1V1+q1V2))/mp.

    In practice, the parameters m, k, βe and Ct may not be known accurately, so it is necessary to consider the uncertainties of these parameters. Define vector as θ=[θ1,θ2,θ3,θ4] and improve the tracking performance of the system through the adaptive method.

    The purpose of the system controller is to design a bounded control input u so that y = x1 can track the desired trajectory yd(t) = x1d(t). Therefore, the following assumptions should be given.

    Assumption 1: The desired trajectory x1d is five times differentiable and each is bounded. In practice, the load pressure of hydraulic cylinder meets 0 < PL < Ps.

    Assumption 2: The range of parametric uncertainties is

    θΩθ (14)
    ΩθΔ{θ:θminθθmax} (15)

    where θmin=[θ1min,,θ4min]T, θmax=[θ1max,,θ4max]T are known.

    Assumption 3: The time-varying perturbation Δ (t) of Eq (10) is smooth enough so that

    ˙Δ(t)∣≤δ1&¨Δ(t)∣≤δ2 (16)

    where δ1, δ2 are known positive constants.

    ESO can estimate the uncertainty disturbance comprehensively, so we use ESO to estimate the lumped disturbance and compensate it feed forward to achieve better tracking accuracy.

    Define x4 = Δ(t), ˙x4=δ(t) and the expanded state space can be written as

    {˙x1=x2˙x2=x3˙x3=θ1uθ2x1θ3x2θ4x3+x4˙x4=δ (17)

    Defining ˆx=xˆx as the estimation error of x, where ˆx represents the estimated value of x.

    According to the expanded state space model, the ESO is designed as

    {˙ˆx1=ˆx2+4ω0(x1ˆx1)˙ˆx2=ˆx3+6ω20(x1ˆx1)˙ˆx3=ˆθ1uˆθ2x1ˆθ3x2ˆθ3x3+ˆx4+4ω30(x1ˆx1)˙ˆx4=ω40(x1ˆx1) (18)

    where ω0 is the bandwidth of the extended state observer, ˆθi is the estimated value of the unknown parameters θi, I = 1, 2, 3, 4.

    The dynamic equation of observation error can be obtained by subtracting Eq (17) an Eq (18):

    {˙˜x1=˜x24ϖ0˜x1˙˜x2=˜x36ω20˜x1˙˜x3=˜x4+˜θTΦ14ω30˜x1˙˜x4=δω40˜x1 (19)

    where ˜θ=[˜θ1,˜θ2,˜θ3,˜θ4]T, Φ1=[u,x1,x2,xξ,x4]T.

    Let

    ξi=˜xiωi10,i=1,2,3,4 (20)

    Then Eq (17) can be written as

    [˙ξ1˙ξ2˙ξ3˙ξ4]=ϖ0[4100601040011000]A[ξ1ξ2ξ3ξ4]+[0010]B1˜θTΦϖ20+[0001]B2δϖ30 (21)
    ˙ξ=ω0Aξ+B1˜θTΦω20+B2δϖ30 (22)

    Since matrix A is a Hurwitz matrix, and there is a positive definite symmetric matrix P which satisfies the following equation

    ATP+PA=I (23)

    The symmetric positive definite matrix P is:

    P=[17/17881/12211/11881/1221/12211/11881/12217/178811/11881/12217/17881/1221/12217/17881/12291/9188]

    Considering Eq (14) and Eq (15), the discontinuous projection can be defined as [11].

    Projˆαi(i)={0,i>0andˆαi=αimax0,i<0andˆαi=αimini,otherwise (24)

    where ˆα denote the estimate of α and ˜α denote the estimate error, ˜α=ˆαα,i=1,2,3,4.

    Using the adaptation law as follow:

    ˙ˆα=Projˆα(Γτ(t)),ˆα(0)=Ωˆα (25)

    where Γ is the diagonal positive definite adaptation rate matrix, τ is an adaptation function. For any adaption function τ, the adaptation Eq (25) satisfies follow [3]:

    ˆαΩ˙α{ˆα:αmaxˆααmin} (26)
    ˜αT[Γ1Projˆα(Γτ)τ]0,τ (27)

    Defining the following error variables

    {z1=x1x1d,z2=˙z1+k1z1z3=˙z2+k2z2,r=˙z3+k3z3 (28)

    where x1d is the given trajectory; k1 k2 k3 are the positive feedback gain and r is the auxiliary error signal. Because r contains the differentiation of acceleration, it is considered to be unmeasurable in practice and only used for auxiliary design. According to Eq (28), r has the following expansion:

    r=θ1uθ2x1 dθ3˙x1 dθ4¨xld+Δxld(θ2k1θ3k31+θ4k21)z1(k21+k1k2k22θ3θ4k1θ4k2)z2+(k1+k2+k3θ4)z3 (29)

    Dividing Eq (29) by θ1, and thus

    α1r=uα1x1 dα2x1 dα3˙xldα4¨x1 d+α1Δ(α2k1α3α1k31+α4k21)z1((k21+k1k2)α1k22α3α4k1α4k2)z2+((k1+k2+k3)α1α4)z3 (30)

    where α1=1/1θ1θ1, α2=θ2/θ2θ1θ1, α3=θ3/θ3θ1θ1, α4=θ4/θ4θ1θ1.

    The model-based controller is designed as follows:

    {u=ua+us,us=(us1+us2),us1=k3z3us2=krz3+krz3(0)to[krk3z3(τ)+βS(z3(τ))]dτua=ˆα1x1d+ˆα2x1d+ˆα3˙x1d+ˆα4¨x1dˆα1ˆΔ=ˆαTΦ2ˆα1ˆΔ (31)

    where ˆα=[ˆα1,ˆα2,ˆα3,ˆα4]T represents the estimated value of α=[α1,α2,α3,α4]T; kr>0 is the gain of controller; β>0 is the robust gain; ua is the feedforward model compensation, us1 is the linear robust feedback term, us2 is the RISE control term, Φ2=[x1d,x1d,˙x1d,¨x1d]T.

    Substituting Eq (31) into Eq (30) and note that d=ˆα1ˆΔ+α1Δ

    α1r=˜αTΦ2+d(k3+kr)z3+krz3(0)to[krk3z3(τ)+βS(z3(τ))]dτAz1Bz2+Cz3 (32)

    ˜α=ˆαα, Substitute Eq (25) into Eq (32), thus

    α1˙r=Projˆα(Γτ)TΦ2+˜αT˙Φ2+˙d(k3+krC)rβS(z3)+Ak1z1(ABk2)z2(B+Ck3)z3 (33)

    The overall structure of the designed control strategy is shown in Figure 2.

    Figure 2.  Overview of control diagram.

    Lemma 1: Define variable L(t) as

    L(t)=r[˙dβsign(z3)] (34)

    Define auxiliary function as

    P(t)=β|z3(0)|z3(0)˙dt0L(v)dv (35)

    According to [8], if the gain β satisfies the following inequality, then the auxiliary function P(t) is always positive definite.

    β (36)

    Theorem 1: Using the adaptive law Eq (25), and adaptive function \tau = - r{{\boldsymbol{\dot \Phi }}_2} , and the robust gain β satisfies inequality Eq (36) as well as the feedback gains k1, k2, k3, kr are sufficient to ensure that the matrix {\Lambda } defined below is positive definite, the adaptive robust integral of the sign of the error controller Eq (31) can make all signals bounded in the closed-loop system, and the system obtains asymptotic output tracking, i.e., {z_1} \to 0 as t \to \infty .

    {\boldsymbol{\varLambda}} = \left[ {\begin{array}{*{20}{c}} {{k_1}}&{ - \frac{1}{2}}&0&{ - \frac{1}{2}{k_5}} \\ { - \frac{1}{2}}&{{k_2}}&{ - \frac{1}{2}}&{ - \frac{1}{2}{k_6}} \\ 0&{ - \frac{1}{2}}&{{k_3}}&{ - \frac{1}{2}{k_{67}}} \\ { - \frac{1}{2}{k_5}}&{ - \frac{1}{2}{k_6}}&{ - \frac{1}{2}{k_7}}&{{k_4}} \end{array}} \right] (37)

    where {k_4} = \max \left({{{{\dot {\boldsymbol{\varPhi}} }}_2}^{\rm T}\boldsymbol{\varGamma }{{\boldsymbol{\varPhi} }_2}} \right) + {k_3} + {k_r} - C , {k_5} = A{k_1} , {k_6} = - \left({A - B{k_2}} \right) , {k_7} = - \left({B + C{k_3}} \right) , \max \left(\cdot \right) represents the maximum value of the matrix.

    Proof: see Appendix A.

    The nominal value of the physical parameters of the valve controlled symmetrical hydraulic cylinder are shows in Table 1. The following controllers are compared by simulation to validate the effectiveness of the designed controller.

    Table 1.  Physical parameters of the valve controlled symmetrical hydraulic cylinder.
    Parameter Value Unit Parameter Value Unit
    mp 0.76167 kg fv 100 N/(m/s)
    Ap 2.5 × 10-4 m2 V10 1 × 10-3 m3
    k 10900 N/m V20 1 × 10-3 m3
    βe 2 × 108 Pa kt 5.656 × 10-8 m3/(s·V·N1/2)
    Ct 1 × 10-13 m3/(Pa·s) q1 1 × 10-12 m3/s
    q2 1 × 10-12 m3/s Cd 0.7
    Cv 1 / Wp 5 e-3
    Δp 821, 993 / α 69 °
    K 1000

     | Show Table
    DownLoad: CSV

    (1) Controller I: ESO based ARISE This is the controller designed in this paper. The controller parameters are selected as: k1 = 200, k2 = 180, k3 = 0.08, kr = 0.012, β = 0.05, ω = 100. According to the nominal value of the parameters, the estimated boundary of unknown parameter α are given as: αmin = [5 × 10-5 0.03 2.5 8 × 10-3] and αmax = [7 × 10-5 0.07 3.5 20.2 × 10-3]. The initial estimates of α is set as \hat \alpha (0) = [6.5×10-5 0.04 3 8.2×10-3] and Γ is set as diag[1 × 10-15 2 × 10-2 0.001 5 × 10-10].

    (2) Controller II: ARISE without ESO. Compared to the controller II, there is no ESO compensation term and the other parameters are same to controller I. That is only uα in Eq (18) is replaced as: {u_a} = {\hat \alpha _1}{\dddot x_{1{\text{d}}}} + {\hat \alpha _2}{x_{1{\text{d}}}} + {\hat \alpha _3}{\dot x_{1{\text{d}}}} + {\hat \alpha _4}{\ddot x_{1{\text{d}}}} = {\hat \alpha ^{\rm T}}{{\boldsymbol{\varPhi }}_2}

    (3) Controller III: PI controller. The parameters are set as kP = 410 and kI = 10, which are the optimal solutions after repeated debugging.

    (4) Controller IV: BP neutral network PID controller. The structure of the neutral network is 3-5-3, and the learning rate \eta = 0.28 , inertia coefficient α = 0.3.

    The desired trajectories are designed as three cases: normal motion with the motion trajectory

    {x_{\text{d}}}\left(t \right) = {{10\arctan \left[{\sin \left({\pi t} \right)} \right]\left({1 - {e^{ - t}}} \right)} \mathord{\left/ {\vphantom {{10\arctan \left[{\sin \left({\pi t} \right)} \right]\left({1 - {e^{ - t}}} \right)} {0.7854}}} \right. } {0.7854}} mm, fast level motion with the motion trajectory {x_{\text{d}}}\left(t \right) = {{10\arctan \left[{\sin \left({4\pi t} \right)} \right]\left({1 - {e^{ - t}}} \right)} \mathord{\left/ {\vphantom {{10\arctan \left[{\sin \left({4\pi t} \right)} \right]\left({1 - {e^{ - t}}} \right)} {0.7854}}} \right. } {0.7854}} mm and low-level motion with the motion trajectory {x_{\text{d}}}\left(t \right) = {{10\arctan \left[{\sin \left({0.2\pi t} \right)} \right]\left({1 - {e^{ - t}}} \right)} \mathord{\left/ {\vphantom {{10\arctan \left[{\sin \left({0.2\pi t} \right)} \right]\left({1 - {e^{ - t}}} \right)} {0.7854}}} \right. } {0.7854}} mm. The external disturbance is designed as f\left(t \right) = {{20\arctan \left[{\sin \left({0.8\pi t} \right)} \right]\left({1 - {e^{ - t}}} \right)} \mathord{\left/ {\vphantom {{20\arctan \left[{\sin \left({0.8\pi t} \right)} \right]\left({1 - {e^{ - t}}} \right)} {0.7854}}} \right. } {0.7854}} N.

    To compare the tracking responses of each controller quantitatively, three performance indices including maximum absolute value of the tracking error Me, average tracking error {\mu _{\text{e}}} , standard deviation of the tracking error σe, which were defined in are adopted to evaluate [20].

    (1) Case I-normal level motion

    The four controllers are tested for a normal motion trajectory xd(t) = 10arctan[sin(πt)] [1-exp(-t)]/0.7854 mm. The tracking performance are shown in Figures 4-6, the performance indices of the four controllers is shown in Table 2. From the Figures 4-6 and Table 1, it is obviously that the valve controlled symmetrical cylinder has the best tracking performance under the controller designed in this paper than other controllers. From Table 2, the amplitudes of steady-state tracking error of the controller III and controller IV are both about 0.6 mm, while controller I is about 0.003 mm and controller II is about 0.01 mm, it shows that the ARISE can deal with nonlinear and uncertainties and disturbance well but PI controller just has some robustness. By comparing the performance indices in Table 2 and tracking error in Figure 6 of controller I and controller II, it can be seen that controller I is better than controller II in all indices obviously, which indicates that the parameter adaptation in Figure 6 and ESO compensates for both parametric and uncertain lumped disturbance are effective. The control input u of controller I showed in Figure 3 is continuous and smooth, which makes it easy to implement in practice.

    Figure 3.  Tracking performance of controller I for normal level motion.
    Figure 4.  Tracking performance of controller I for normal level motion.
    Figure 5.  Tracking errors of four controllers I for normal level motion.
    Figure 6.  Parameter adaptation of controller I.
    Table 2.  Performance indices during the last two cycles for normal motion case.
    Indices Me (mm) μe (mm) σe (mm)
    controller I 0.00304985 0.000778897 0.00084379
    controller II 0.010971 0.00364635 0.00293964
    controller III 0.63097 0.299917 0.197092
    controller IV 0.630986 0.299916 0.197092

     | Show Table
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    (2) Case II-low level motion

    In this case, a low level reference trajectory xd(t) = 10arctan[sin(0.2πt)][1-exp(-t)]/0.7854 mm is tested. The results are shown in Figures 7-9 and the performance indices are listed in Table 3. From Figure 7 and Table 3, the amplitudes of steady-state tracking error of the controller III and controller IV are both about 0.076mm, while controller I and controller II both are about 9.5 × 10-5 mm, which shows that the ARISE can also deal with nonlinear and uncertainties and disturbance well in low level reference trajectory. By comparing the performance indices of controller I and controller II in Table 3, it can be seen that the maximum absolute value of the tracking error Me of the two controllers almost the same, but the average tracking error μe, and standard deviation of the tracking error σe of controller I are 6.536 × 10-6 mm and 8.833 × 10-6 mm respectively which are better than that of 2.87371 × 10-5 and 1.02822 × 10-5 of controller II obviously, which further validates the effectiveness of the desired parameter adaptation and ESO compensation. The parameter adaptation of controller I are omitted. The control input u of controller I showed in Figure 8 is continuous and there is slight high-frequency vibration. So the controller designed in this paper has the best tracking performance on low level reference trajectory than other controllers too.

    Figure 7.  Tracking performance of controller I for low level motion.
    Table 3.  Performance indices during the last two cycles for slow motion case.
    Indices Me(mm) μe(mm) σe(mm)
    controller I 9.508 × 10-5 6.536 × 10-6 8.833 × 10-6
    controller II 9.58784 × 10-5 2.87371 × 10-5 1.02822 × 10-5
    controller III 7.5716 × 10-2 3.14012 × 10-2 2.00589 × 10-2
    controller IV 7.57345 × 10-2 3.13926 × 10-2 2.00719 × 10-2

     | Show Table
    DownLoad: CSV
    Figure 8.  Control input of controller I for normal level motion.
    Figure 9.  Tracking errors of four controllers I for low level motion.

    (3) Case III-fast level motion

    A faster level reference trajectory xd(t) = 10arctan[sin(4πt)][1-exp(-t)]/0.7854 mm is tested in this case. The results are shown in Figured 10-12 and the performance indices are listed in Table 4. From Figure 12 and Table 4, the amplitudes of steady-state tracking error of the controller III and controller IV are both about 1.24 mm, while controller I 1.34 × 10-2 mm and controller II 4.20256 × 10-2 mm, which shows that the ARISE can better deal with nonlinear and uncertainties and disturbance in fast level reference trajectory than PI controller too. In addition, comparing the performance indices of controller I and controller II in Table 4, it can be seen that the maximum absolute value of the average tracking error μe of the controller I is 4.14 × 10-3 mm and the controller II is 1.63032 × 10-2 mm, the standard deviation of the tracking error σe of controller I are 5.53 ×10-3 mm and the controller II is 1.30355 × 10-2 mm. These further validate the effectiveness of the desired parameter adaptation and ESO compensation. The control input u of controller I showed in Figure 11 is continuous and smooth. The result verifies that the controller designed in this paper still has high tracking accuracy in tracking performance on fast level reference trajectory.

    Figure 10.  Tracking performance of control-ler I for fast level motion.
    Figure 11.  Control input of controller I for normal level motion.
    Figure 12.  Tracking errors of four controllers I for fast level motion.
    Table 4.  Performance indices during the last two cycles for fast motion case.
    Indices Me(mm) μe(mm) σe(mm)
    controller I 1.34 × 10-2 4.14 × 10-3 5.53 × 10-3
    controller II 4.20256 × 10-2 1.63032 × 10-2 1.30355 × 10-2
    controller III 1.238558 0.598359 0.391225
    controller IV 1.238657 0.598359 0.391224

     | Show Table
    DownLoad: CSV

    In this paper, an ARISE with ESO controller is proposed for EHSS to address parametric uncertainties, uncertainty nonlinearities and unmolded disturbances. The proposed ARISE can compensate the dynamics uncertainties, thus guaranteeing asymptotic tracking and improving the adaptability and safety of EHSS. ESO can effectively estimate the state and lumped uncurtains. Simulation results shows that ARISE with ESO can obtain high tracking accuracy and better performance in tracking desired trajectory under all working conditions.

    The authors declare there is no conflict of interest.

    Define a Lyapunov function

    V = \frac{1}{2} z_1^2+\frac{1}{2} z_2^2+\frac{1}{2} z_3^2+\frac{1}{2} \alpha_1 r^2+\frac{1}{2} \tilde{\boldsymbol{\alpha}}^{\mathrm{T}} \boldsymbol{\varGamma}^{-1} \tilde{\boldsymbol{\alpha}}+P (A.1)

    It is Obvious that V is positive definite. The derivative of V is:

    \dot{V} = z_1 \dot{z}_1+z_2 \dot{z}_2+z_3 \dot{z}_3+\alpha_1 r \dot{\eta}+\tilde{\boldsymbol{\alpha}}^{\mathrm{T}} \boldsymbol{\varGamma}^{-1} \dot{\hat{\boldsymbol{\alpha}}}+\dot{P} (A.2)

    Substituting Eqs (25) and (28), L(t)and P(t), into Eq (A.2).

    \begin{aligned} \dot{V} & = z_1\left(z_2-k_1 z_1\right)+z_2\left(z_3-k_2 z_2\right)+z_3\left(r-k_3 z_3\right)+ \\ & r\left\{-\dot{\boldsymbol{\varPhi}}_2^{\mathrm{T}} \boldsymbol{\varGamma} \boldsymbol{\Phi}_2 r+\tilde{\boldsymbol{\alpha}}^{\mathrm{T}} \dot{\boldsymbol{\varPhi}}_2+\dot{d}-\left(k_3+k_r-C\right) r-\beta \operatorname{sign}\left(z_3\right)+A k_1 z_1-\left(A-B k_2\right) z_2-\left(B+C k_3\right) z_3\right\} \\ & +\tilde{\boldsymbol{\alpha}}^T \boldsymbol{\varGamma}^{-1} \operatorname{Proj}_{\hat{\theta}}(\Gamma \tau)-r\left[\dot{d}-\beta \operatorname{sign}\left(z_3\right)\right] \\ & \leq z_1\left(z_2-k_1 z_1\right)+z_2\left(z_3-k_2 z_2\right)+z_3\left(r-k_3 z_3\right)+ \\ & r\left\{-\dot{\boldsymbol{\varPhi}}_2^{\mathrm{T}} \boldsymbol{\varGamma} \boldsymbol{\Phi}_2 r+\tilde{\boldsymbol{\alpha}}^{\mathrm{T}} \dot{\boldsymbol{\varPhi}}_2+\dot{d}-\left(k_3+k_r-C\right) r-\beta \operatorname{sign}\left(z_3\right)+A k_1 z_1-\left(A-B k_2\right) z_2-\left(B+C k_3\right) z_3\right\} \\ & +\tilde{\boldsymbol{\alpha}}^T \tau-r\left[\dot{d}-\beta \operatorname{sign}\left(z_3\right)\right] \\ & = -k_1 z_1^2-k_2 z_2^2-k_3 z_3^2-\left(\dot{\boldsymbol{\varPhi}}_2^{\mathrm{T}} \boldsymbol{\varGamma} \boldsymbol{\Phi}_2+k_3+k_r-C\right) r^2+z_1 z_2+z_2 z_3+z_3 r+\left(A k_1 z_1-A z_2\right) r+B k_2 z_2 r-\left(B+C k_3\right) z_3 r \\ & \triangleq-\boldsymbol{\eta}^{\mathrm{T}} \boldsymbol{\varLambda} \boldsymbol{\eta} \leq-\lambda_{\min }(\boldsymbol{\varLambda})\left(z_1^2+z_2^2+z_3^2+r^2\right) \triangleq-W \end{aligned} (A.3)

    where \boldsymbol{\eta} = \left[z_1, z_2, z_3, r\right]^{\mathrm{T}}, \lambda \min (\boldsymbol{\varLambda}) is the minimum eigenvalue of matrix Λ, therefore V \in L_{\infty} and W \in L_2, so z1, z2, z3 and r are bounded. According to the assumptions 1 and 2, all states of the system are bounded so the actual control input u is bounded. According to Eqs (28) and (33), the derivative of W is bounded, so W is uniformly continuous. According to Barbarat's lemma, W→0 as t→∞, so the conclusion of the theorem 1 can be deduced, theorem 1 is proofed.



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