# Adaptive Neural Network Control for Nonlinear Hydraulic Servo-System with Time-Varying State Constraints

Authors: Shu-Min Lu; Dong-Juan Li
Journal: Complexity
 (2017)

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## Abstract

An adaptive neural network control problem is addressed for a class of nonlinear hydraulic servo-systems with time-varying state constraints. In view of the low precision problem of the traditional hydraulic servo-system which is caused by the tracking errors surpassing appropriate bound, the previous works have shown that the constraint for the system is a good way to solve the low precision problem. Meanwhile, compared with constant constraints, the time-varying state constraints are more general in the actual systems. Therefore, when the states of the system are forced to obey bounded time-varying constraint conditions, the high precision tracking performance of the system can be easily realized. In order to achieve this goal, the time-varying barrier Lyapunov function (TVBLF) is used to prevent the states from violating time-varying constraints. By the backstepping design, the adaptive controller will be obtained. A radial basis function neural network (RBFNN) is used to estimate the uncertainties. Based on analyzing the stability of the hydraulic servo-system, we show that the error signals are bounded in the compacts sets; the time-varying state constrains are never violated and all singles of the hydraulic servo-system are bounded. The simulation and experimental results show that the tracking accuracy of system is improved and the controller has fast tracking ability and strong robustness.


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## Body

## 1. Introduction

Throughout history, the hydraulic servo-system has been always widely applied in many aspects, such as aerospace, aviation, navigation, weapon, mining, and metallurgy due to the advantages of high response, high power, large stiffness, strong robustness capability, small volume, and so on. However, serious nonlinear behavior, such as control input saturation [1], state constraint, valve opening, nonlinear friction [2], and model uncertainty [3] (load changes, the parameters variation and the element parameter uncertainty [4] caused by the abrasive, containing external disturbance [5, 6], leakage, and other uncertain nonlinear elements), restricts the development of high-performance closed loop system controller [7, 8].Recently, in order to solve the uncertain nonlinear problem in the system, the most authors adopted the fuzzy logical system or adaptive neural network (NN) control [9, 10] to approximate the external disturbance [11] and model uncertainty [12, 13]. In [14, 15], the different stability problems of switched nonlinear systems are solved by using the fuzzy adaptive control. The paper [14] solved the discrete-time switch system tracking control problem, and [15, 16] presented decentralized controller for switched uncertain nonlinear large-scale systems with dead zones based on observer [17]. The article [18] solved the control problem of the nonlinear system with dynamic uncertainties and input dead zone. Similarly, the authors in [19] used the NN to deal with the uncertain problem of nonlinear nonstrict feedback discrete-time systems [20]. The article [21] adopted general projection neural network (GPN) to iteratively solve a quadratic programming (QP) problem, which updates the algorithm of NN to develop the new application. Two neural networks which include the critic NN and the action NN are used to approximate the strategic utility function and to minimize both the strategic utility function and the tracking error in [22], respectively. The articles [23–26] adopt the adaptive NN in quantized nonlinear systems, nonlinear time-delay systems, and uncertain nonlinear systems. These articles extend the theoretical application of neural network. In [27], an adaptive neural control scheme is presented to take the unknown output hysteresis and computational efficiency into account. The motion/force performances limits of the robotic system will be relaxed. The NN turns out to be a clever theory to solve the uncertain information of the systems from the above articles.At present, the control problem of hydraulic servo-system has been an active research field by designing the adaptive control technique. The authors in [28] have presented an adaptive controller based on the robust integral of the error signal to deal with modeling uncertainties of the system, and the feedback gain is reduced. In [29], the robust adaptive control ensures uniform boundedness of the thruster assisted position mooring system which is in the transverse direction under the ocean current disturbance. The robust adaptive controller has been designed for single-rod nonlinear hydraulic servo-system where the element parameter uncertainty and nonlinear uncertainty behavior exist in [30]. The article in [31] presents output feedback nonlinear robust control based on an extended state observer (ESO) to solve mismatched modeling uncertainties problem, which is very important for high-accuracy tracking control of hydraulic servo-systems. In [32], considering the influence of friction on the system, an adaptive controller is designed to solve the problem of system friction and parameter uncertainty. In fact, due to the high bearing capacity and high stiffness properties of the hydraulic servo-system, so the state constraints problems are ignored in the environment and the interaction of measurement unit tests process, which lead to too large displacement, velocity, or acceleration causing damage to the measuring equipment during testing process [33]. In tests and experiments, if the initial conditions of the system do not match, it is possible that the displacement, velocity, or acceleration of the system is oversized. At the same time, the constraint for the system is a good way to achieve high accuracy tracking performance. However, none of the above articles considered constraints condition. And considering that the state variables vary with time, the constraint condition should also be considered the factor of time changing. Meanwhile, compared with constant constraints, the time-varying state constraints are more general in the actual systems.In fact, some systems are often affected by constraints [34], such as temperature constraints in chemical reactions and the speed or acceleration constraints when some mechanical systems suffer physical failure [35]. The authors first present the theory of nonlinear system with output constrains and time-varying output constrains by using Barrier Lyapunov in [36, 37]. However, those articles do not consider the interference of unknown functions. In [38], considering a class of special nonlinear systems with the hysteretic output mechanism and the unmeasured states, adaptive neural output feedback control is presented to guarantee the prescribed convergence of the tracking error [39]. The article in [40] has studied adaptive neural control of uncertain stochastic nonlinear systems with dead zone and output constraint. The continuous stirred tank reactor [41], flexible crane system [42], and robot system [43, 44] all design adaptive NN controller based on output constrains condition. It is essential that the theories of constraint are combined with the actual systems [45]. Similarly, the state constrains have also been researched by many scientific researchers. For example, the article in [46] presents adaptive fuzzy NN control using impedance learning for a constrained robot. The authors considering time-delay condition investigate the robotic manipulator system [47], nonlinear MIMO unknown systems [48], and common nonlinear systems [49] with full state constrains [50]. The authors further research the nonlinear pure-feedback systems, Nussbaum gain adaptive control, and ABLF adaptive controller to apply them on stochastic nonlinear systems and wheeled mobile robotic system by using the theory of integral barrier Lyapunov function or state constrains in [51], [52], [53], [54], [55], and [56], respectively.Based on the above descriptions, this article presents an adaptive NN tracking control for hydraulic servo-system based on time-varying state constraint, and hydraulic system adaptive control is first designed with the time-varying state constraints. Finally, when the states of the system are forced to obey bounded time-varying constraint conditions, the high precision tracking performance of the system can be easily realized. Meanwhile, it can be proved that the tracking error signal converges to zero asymptotically and all singles of the hydraulic servo-system are bounded by using the Lyapunov analysis. The experimental results show the availability of the design controller.


## 2. System Descriptions and Problem Preliminaries

A class of hydraulic servo-system dynamics of the inertial load can be described as the following form:(1)mx¨=PeA-ρx˙-dx,x˙,t,where x represents the displacement of the inertial load; m is the mass of moving parts; Pe=P1-P2, P1 and P2 are pressures of the left and right inside the hydraulic cylinder, respectively; A is the corresponding areas of the above-mentioned chamber; ρ represents the viscous friction coefficient; dx,x˙,t is defined as the unconsidered disturbance.Considering the compressibility of the hydraulic oil and ignoring leakage of hydraulic cylinder, the pressure dynamic equation of both chambers is as follows:(2)Ve4BvP˙e=-Ax˙-CmPe-w+Q,where Ve=Vs1+Vs2 is the total effective cylinder volume; Vs1=VI1+Ax and Vs2=VI2-Ax represent the total volume of the left and the right chamber inside hydraulic cylinder, respectively; VI1 and VI2 are the corresponding the initial volume; Bv is the effective bulk modulus of hydraulic cylinder; Cm represents the total internal leakage coefficient of hydraulic cylinder; w represents the modeling error of both chambers; Q=Q1+Q2/2 is the load-flow of hydraulic cylinder; Q1 and Q2, respectively, represent the supplied oil flow rate to the left chamber and the returned oil flow rate to the right chamber.Consider that the flow ratesQ1 and Q2 are the function of the spool displacement xs. The flow equation of hydraulic cylinder is written as follows:(3)Q1=kfxsh1xs,P1,Q2=kfxsh2xs,P2with(4)h1xs,P1=Ps-P11/2,xs≥0,P1-Pr1/2,xs<0,h2xs,P2=P2-Pr1/2,xs≥0,Ps-P21/2,xs<0,where kf denotes the flow gain coefficient; Ps is the supply pressure of the oil; Pr represents the return oil pressure.We assume that the spool displacementxs is direct proportion with control input voltage u in the situation of ignoring the high frequency; that is, xs=ksu, ks>0 is the small gain. Then, (3) can be rewritten as(5)Q1=Kfuh1u,P1,Q2=Kfuh2u,P2,where Kf=kfks represents the total flow gain coefficient. Then, we can conclude that the load-flow of hydraulic cylinder is(6)Q=KfuPs-sgn⁡uPe1/2,where sgnu represents the step function as(7)sgn⁡u=1,u≥0,-1,u<0,Ps=P1+P2,Pr=0,Pe=P1-P2.Define state variablesx=x,x˙,x¨T=x1,x2,x3T; the dynamics of the hydraulic servo-system (1) can be translated into the following the state space expression based on the pressure dynamic equation (2) and the flow equations (5)-(6):(8)x˙1=x2,x˙2=x3,x˙3=-f1x2-f2x+f3gu,where(9)f1=4A2BvmVe-ρ2m,f3=4ABvKfmVe,g=Ps-sgn⁡uPe2f2x=4ABvCmPemVe+4ABvwmVe+ρAPem2-ρmdx+d˙xm.f1x, f2x, and f3x are unknown smooth parameter functions. In this paper, the state variable xi is constrained in the following time-varying compact sets:(10)xit≤kbit,i=1,2,3,where kbit represents the time-varying smooth functions.Assumption 1 (see [37]).
There exist the functionsf-3x and f_3x which satisfy the inequality 0<f_3≤f3≤f-3.Assumption 2 (see [43]).
There are functionsY0t, Y1t, and Y2t. We assume that the functions satisfy the inequalities Yi-1t≤kbit,i=1,2,3. Then, the smooth desired trajectory ydt and its time derivatives, respectively, satisfy ydt≤Y0t and ydit≤Yi+1t,i=1,2 for all t>0.Lemma 3 (see [54]).
For any time-varying functionkat, the following inequality holds for all error function zt in the interval zt≤kat:(11)log⁡ka2tka2t-z2t≤z2tka2t-z2t.The above-mentioned unknown smooth functionsf1, f2x, and f3 are approximated by the radial basis function NN as(12)fiX=WiTΦiX,i=1,2,3,where Wi∈Rli denotes the neural network weight vector and ΦiX=φi,1,…,φi,li∈Rli are the basis function vectors with input vector X∈Rni and the NN node number li>1. Gaussian functions φi,jX are chosen in the following forms:(13)φi,jX=exp⁡-X-μi,j2ηi,j2,i=1,2,3,j=1,…,li,where μi,j=πi,j,1,…,πi,j,niT denotes the center of NN function and ηi,j is the width of the Gaussian function.Define the unknown optimal weight vector as the following form:(14)Wi∗=argminWi∈Rli⁡supX∈Rni⁡WiTΦiX-fiX,i=1,2,3.Then, there exists the unknown approximation errors εiX which has supremum ε-i; that is, εiX≤ε-i. Equation (12) will be rewritten in the approximation equations as follows:(15)fiX=Wi∗TΦiX+εiX,i=1,2,3.


## 3. The Adaptive NN Controller Design

Define the track trajectoryyd and the virtual controllers α1 and α2; we have the track error z1=x1-yd. Then, let z2=x2-α1 and z3=x3-α2. Choose the barrier Lyapunov function candidate as(16)V=∑i=13Vi+VN,Vi=12log⁡kai2kai2-zi2,VN=12∑j=13W~jTΘj-1W~j,where log⁡· represents the natural logarithm; kait=kbit-Dit is boundaries of error vector zi from subsequent feasibility analysis; Θj,j=1,2,3 are the positive gain matrices; W~i is the neural network weight error with W~i=W^i-Wi∗, and note that V is the positive definite and continuously differentiable in the set zit≤kait for i=1,2,3.Based on (16), the time derivative of V is(17)V˙=∑i=13V˙i+V˙N=∑i=13zikai2-zi2z˙i-k˙aikaizi+∑j=13W~jTΘj-1W~˙j.Design the virtual controllers as(18)α1=-γ1+γ-1tz1+y˙d,(19)α2=-γ2+γ-2tz2+α˙1-ka22-z22ka12-z12z1,where the time-varying gains are given by(20)γ-it=k˙aikai2+δi,i=1,2,3for all positive constants δi and γi. When any k˙ai is zero, δi will ensure that the time derivatives of virtual controllers αi and controller u are bounded.Substituting (8) and (18)–(20) into (17), we can obtain(21)V˙=-∑i=12γi+γ-it+k˙aikaizi2kai2-zi2+z2z3ka22-z22+z3ka32-z32x˙3-α˙2-k˙a3ka3z3+∑j=13W~jTΘj-1W~˙j,where(22)α˙1=∂α1∂x1x˙1+∑k=01∂α1ydkydk+1+∂α1ka1kka1k+1,α˙2=∑j=12∂α2∂xjx˙j+∑k=02∂α2ydkydk+1+∂α2ka1kka1k+1+∂α2ka2kka2k+1.According to approximation equations (15), (21) becomes(23)V˙=-∑i=12γi+γ-it+k˙aikaizi2kai2-zi2+∑j=13W~jTΘj-1W^˙j+z3ka32-z32-W1∗TΦ1Xx2-W2∗TΦ2X+W3∗TΦ3Xgu+Dx,t-α˙2+ka32-z32ka22-z22z2-k˙a3ka3z3,where Dx,t represents the systematic disturbance compensation as the following form:(24)Dx,t=-ε1x2-ε2+ε3gu.Choose the controller as follows:(25)u=1W^3TΦ3g-γ3+γ-3tz3+W^1TΦ1x2+W^2TΦ2x+α˙2-ka32-z32ka22-z22z2-z32ka32-z32.Adopting the equationsWi∗=W^i-W~i,i=1,2,3 and controller (25), the expression for (23) can be rewritten by the above controller as(26)V˙=-∑i=13γi+γ-it+k˙aikaizi2kai2-zi2+W~1TΘ1-1W^˙1+Φ1Xx2z3ka32-z32+W~2TΘ2-1W^˙2+Φ2Xz3ka32-z32+W~3TΘ3-1W^˙3-Φ3Xz3guka32-z32+Dx,tz3ka32-z32-z322ka32-z322.Based on the above-mentioned definition ofγ-it, we can obtain that(27)γ-it+k˙aikai≥0,i=1,2,3.Introducing the projection algorithm of adaptive control, we design the adaptive law from (26) as follows:(28)W^˙1=-Θ1Φ1Xx2z3ka32-z32+σ1W^1,Θ1≤0,W^1>Mf1,Pf1W^1,Φ1,Θ1>0,W^1=Mf1,where Pf1W^1,Φ1 represents projection operator as the following form:(29)Pf1W^1,Φ1=-Θ1Φ1Xx2z3ka32-z32-Φ1XW^1TW^1x2z3W^12ka32-z32+σ1W^1.Mf1 is the minuteness positive constant to ensure the adaptive law W^˙1≥0 in any situation.Similarly, the other adaptive laws can obtain the forms as(30)W^˙2=-Θ2Φ2Xz3ka32-z32+σ2W^2,Θ2≤0,W^2>Mf2,Pf2W^2,Φ2,Θ2>0,W^2=Mf2,(31)Pf2W^2,Φ2=-Θ2Φ2Xz3ka32-z32-Φ2XW^2TW^2z3W^22ka32-z32+σ2W^2,(32)W^˙3=Θ3Φ3Xz3guka32-z32-σ3W^3,Θ3≤0,W^3>Mf3,Pf3W^3,Φ3,Θ3>0,W^3=Mf3,(33)Pf3W^3,Φ3=-Θ3Φ3Xz3guka32-z32-Φ3XW^3TW^3z3guW^32ka32-z32+σ3W^3.In particular, when componentW^3,i of W^3 is a small constant mf3,i, that is, W^3=Mf3, we adopt the following component of adaptive law:(34)W^˙3,i=-Θ3,iΦ3,iXz3guka32-z32+σ3W^3,i,0.Remark 4.
Based on the above description (34), we can make the following conclusion. If W^3,i=mf3,i, we can get W^˙3,i>0, that is, W^3,i≥mf3,i>0 for all any W^3,i. Then, there does not exist insignificance function for controller u.Substituting (28), (30), and (32) into (26), we obtain(35)V˙≤-∑i=13γi+γ-it+k˙aikaizi2kai2-zi2-∑i=13σiW~iTW^i+Dx,tz3ka32-z32-z322ka32-z322.Using Young’s inequality in the last two terms ofV˙, we seem to easily get(36)-σiW~iTW^i=-σiW~iTW~i+Wi∗≤-σiW~i2+σi2-Wi∗2+σi2W~i2=-σi2W~i2+σi2Wi∗2,Dx,tz3ka32-z32≤12D2x,t+z322ka32-z322.Based on definition (27) and using inequalities (36) and (11) in Lemma 3, (35) can be rewritten as(37)V˙≤-∑i=13γilog⁡kai2kai2-zi2-∑i=13σi2W~i2+12D2x,t+∑i=13σi2Wi∗2.Theorem 5.
A class of hydraulic servo-system is described by (1)–(3) under Assumptions 1 and 2. If the initial displacement x10, the initial velocity x20, and the initial acceleration x30 satisfy xi0≤kbi0,i=1,2,3, then, the proposed control method has the following properties.
(1) The error signals are bounded in the following sets:(38)zit∈Ωzi,i=1,2,3,where Ωzi≔zi∈R:zit≤kait1-e-V0e-ψt-C/ψ, noting that the tracking error signal converges to zero asymptotically.
(2) The time-varying state constrains are never violated; that is,xit≤kbit,∀t≥0,i=1,2,3.
(3) All singles of the hydraulic servo-system are bounded.Proof.
Please see the Appendix.


## 4. Simulation Example

Simulation has been performed to demonstrate the performance of the proposed approach. The values of system parameters which are used in the simulation are given in Table1.Table 1
The system parameters used in the simulation.
Name
Value
m
k
g
2200
P
s
P
a
20
P
r
P
a
0
A
m
3
2
×
1
0
-
3
ρ
N
s
/
m
500
V
I
1
m
3
5
×
1
0
-
4
V
I
2
m
3
5
×
1
0
-
3
B
v
P
a
200
C
m
m
5
/
N
s
2
×
1
0
-
5
K
f
m
3
/
N
s
8
×
1
0
-
2The initial states of the hydraulic servo-system are given asx10=0, x10=0, and x10=0. The time-varying constraint functions are kb1t=2sin⁡5t+40, kb2t=2sin⁡5t+60, and kb3t=4sin⁡(5t)+150. The desired trajectory tracking periodic reciprocation of the displacement is given yd=40arctan⁡sin⁡0.4πt1-e-t with t∈0,tf and tf=50s. The modeling error disturbance of the both chambers is w=sin⁡t+2. We choose the following controller:(39)α1=-γ1+γ-1tz1+y˙d,α2=-γ2+γ-2tz2+α˙1-ka22-z22ka12-z12z1,u=1W^3TΦ3g-γ3+γ-3tz3+W^1TΦ1x2+W^2TΦ2+α˙2-ka32-z32ka22-z22z2-z32ka32-z32,where we choose design parameters as Y0=20, Y1=10, Y2=50, γ1=10, γ2=3, γ3=6, δ1=0.8, δ2=0.7, δ3=0.6 the initial weight vector estimation W^1=-100, W^2=20, W^3=10, Θ1=3Θ2=2, Θ3=2, σ1=2, σ2=0.3, σ3=0.5, Φ1=0.2, Φ2=0.4, Φ3=0.06.Figures1–6 are illustrated to show the simulation results. Figures 1–3 show the better tracking trajectories performance of the system variable state and minor error curve of the errors on the time-varying constraints, respectively. It can be easily obtained that the time-varying state constrains are not violated. In Figure 4, we can clearly get the conclusion that the tracking errors are all bounded and the tracking error signal curve converges to zero. Figures 5 and 6 show the trajectories of estimation and the controller, respectively. We make the conclusion that the adaptive laws and the controller of the system are all ensured boundedness.Figure 1
(a) Trajectories ofx1 (dashed) and yd (solid); (b) the trajectory of tracking error z1.
(a)
(b)Figure 2
(a) Trajectories ofx2 (dashed) and y˙d (solid); (b) the trajectory of tracking error z2.
(a)
(b)Figure 3
(a) Trajectories ofx3 (dashed) and y¨d (solid); (b) the trajectory of tracking error z3.
(a)
(b)Figure 4
The phase portrait of tracking errorsz1, z1, and z3.Figure 5
(a) The trajectory ofW^1; (b) the trajectory of W^2; (c) the trajectory of W^3.
(a)
(b)
(c)Figure 6
The trajectory ofu.


## 5. Conclusion

An adaptive control algorithm for hydraulic servo-system considering time-varying state constraints has been designed in this paper. The uncertainty and time-varying state constraints problem of the system have been considered in the controller designing process. The uncertainties and the accurate estimation of disturbance have been solved by RBFNN. By designing barrier Lyapunov function to solve the time-varying state constraints problem of system, the stability of the control strategy has been proved. Besides, the proposed method has proved that the error signals are bounded in the small compacts sets, noting that the tracking error signal converges to zero asymptotically; the time-varying state constrains are never violated and all singles of the hydraulic servo-system are bounded. The experimental results have showed the effectiveness of the proposed method.



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