Coupled-disturbance-observer-based position tracking control for a cascade electro-hydraulic system

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Research article

Coupled-disturbance-observer-based position tracking control for a cascade electro-hydraulic system Qing Guo a,n, Jing-min Yin a, Tian Yu b, Dan Jiang c a

School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, 611731 Chengdu, China Center for Power Transmission and Motion Control, Department of Mechanical Engineering, University of Bath, BA2 7AY Bath, UK c School of Mechatronics Engineering, University of Electronic Science and Technology of China, 611731 Chengdu, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 17 June 2016 Received in revised form 11 February 2017 Accepted 21 February 2017

The disturbance suppression is one of the most common control problems in electro-hydraulic systems. especially largely an unknown disturbance often obviously degrades the dynamic performance by biasing the desired actuator outputs (e.g., load forces or torques). In order to reject the dynamic disturbances in some multi-degree-of-freedom manipulators driven by electro-hydraulic actuators, this paper proposes a state feedback control of the cascade electro-hydraulic system based on a coupled disturbance observer with backstepping. The coupled disturbance observer is designed to estimate both the independent element and the coupled element of the external loads on each electro-hydraulic actuator. The cascade controller has the ability to compensate for the disturbance estimating, as well as guarantees the system state error convergence to a prescribed steady state level. The effectiveness of the proposed controller for the suppression of largely unknown disturbances has been demonstrated by comparative study, which implies the proposed approach can achieve better dynamic performance on the motion control of Two-Degree-of-Freedom robotic arm. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Electro-hydraulic system Cascade controller Coupled disturbance observer Two-Degree-of-Freedom robotic arm

1. Introduction Electro-hydraulic servo systems (EHSs) are a hydraulic transmission system, where servo valve (control element) usually throttles oil to control the position/force of hydraulic cylinder (actuator) to drive the motion control of mechatronic plant. EHS is widely used in actual engineering due to their sufficient power for large external load, although its mechanical efficiency and fast response performance is not higher than series elastic actuator (SEA) [1] and shape memory alloy (SMA) actuator [2]. Fales [3] proposed an H∞ feedback linearization method in hydraulic wheel loader. Yao [4] investigated the position coupling problem of electro-hydraulic load simulator by nonlinear optimal compensation control strategy. Subsequently, the insulator fatigue test device driven by electro-hydraulic system is built to verify the fractional order control method by Zhao [5]. Guo [6] considered the electro-hydraulic actuator in the gait switch control of exoskeleton. Shen [7] studied the vibration excitation and force loading n Correspondence to: School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, No. 2006, Xiyuan Ave, West Hi-Tech Zone, Chengdu 611731, China. E-mail addresses: [email protected] (Q. Guo), [email protected] (J.-m. Yin), [email protected] (T. Yu), [email protected] (D. Jiang).

on electro-hydraulic hybrid system. However, there exist two fundamental problems to be concerned in hydraulic control, which are discussed as follows. One fundamental problem is that the high-quality dynamic behavior of EHS cannot be always maintained, since some parametric uncertainties are ignored in many output or state feedback controllers. These parametric uncertainties include unknown parameters and uncertain nonlinearities [8], which are caused by unknown viscous damping, load stiffness, variations in control fluid volumes, physical characteristics of valve, bulk modulus and oil temperature variations [9]. Although linear control methods have been used in EHS such as PID [10], linear discrete controller [11], some nonlinear dynamic behaviors may be neglected due to local linearization. Then many advanced controllers have been presented to overcome parametric uncertainties of EHS such as robust H∞ controller [12,13], quantitative feedback control scheme [14], geometric control approach [15], output regulation control [16], backstepping control [17,21,22], parametric adaptive controller [18–20]. These nonlinear controllers have been proved to efficiently improve the robustness of hydraulic control. The other problem in EHS is undesirable dynamic behaviour of the established controller, due to the ignorance of largely unknown disturbance caused by dynamic external load (i.e., torque/ force). As far as the authors are concerned, the external load was not well addressed in aforementioned studies. If the external load

http://dx.doi.org/10.1016/j.isatra.2017.02.014 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Guo Q, et al. Coupled-disturbance-observer-based position tracking control for a cascade electro-hydraulic system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.014i

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Nomenclature Two-DOFTwo-degree-of-freedom EHS Electro-hydraulic system EHA Electro-hydraulic actuator CDOB Coupled disturbance observer Ksvi, Tsvi Gain and time constant of the i th servo valve ui Input control voltage of the i th servo valve Cd Discharge coefficient pLi, ps Load pressure of the i th servo valve and supply pressure xvi Spool position of the i th servo valve

disturbance is considered as zero in [21] or a known value in [22], or a boundary uncertainty in [23], many novel controllers are designed conveniently by Lyapunov technique to address these load disturbances. Yao and Bu [24,25] assumed that the maximum relative uncertainty of the external load disturbance was bounded by a known value and proposed a discontinuous projection-based adaptive backstepping controller. Yao [26] presented an adaptive robust controller to handle the nonlinear parametric uncertainty in an auxiliary function, and assumed that the external load disturbance became zero after a finite time in [27]. Kaddissi [28] proposed an equivalent parameter identification method by the least squares and obtained better performance than linear controller, although the load torque disturbance was still assumed as zero. However, in engineering practice, the external load is often largely unknown disturbance in EHS. The designed controller would not only eliminates system state error but also suppresses unknown disturbance of external load. Even though the EHS may be stable, the dynamic performance will be declined if the external load increases beyond a definite boundary [29,30]. Subsequently, Pan [31,32] proposed an adaptive dynamic surface control in a class of strict-feedback nonlinear systems with parametric uncertainties. Then Xu [33,34] adopted dynamic surface control in the longitudinal motion of hypersonic flight vehicle. He and Zhang [35–38] used adaptive neural network in robotic manipulator with input and output constraints. Kim [40] proposed a flatness-based nonlinear controller to improve the position tracking performance while assuming the known constant external load. Then his research team developed a high-gain disturbance observer (HDOB) with backstepping control to compensate for the unknown external load and guaranteed the position tracking accuracy. In this novel approach, the disturbance observer had two different forms. One was a second order high-pass filter [41] to estimate a sinusoidal disturbance with unknown frequency. The other was a HDOB [42] to estimate largely unknown disturbance caused by the friction, the load force, and the parameter uncertainties. However, in some multi-dof manipulator driven by more than two electro-hydraulic actuators such as crane, robotic arm, and exoskeleton, the external loads on respective electrohydraulic actuators (EHAs) are often intercoupled each other. The coupled disturbances from multi-dof manipulator become more complicated than that of one fundamental EHS, which will degrade the estimated accuracy of the disturbance observer and the dynamic response performance of the EHS. In this study, a backstepping cascade controller with a coupleddisturbance observer is proposed to handle largely unknown disturbance on each EHA of multi-dof manipulator. This coupleddisturbance is a typical issue in multi-dof manipulator. Different from one-degree-of-freedom plant, the external load disturbances on several EHAs are more complex without definite frequency due to the coupled motion. Thus, the general high-gain disturbance

w ρ Ctl βe Ap Vt mi, K b FLi yid, yi sgn(.) tanh(.)

Area gradient of the servo valve spool Density of hydraulic oil Coefficient of the total leakage of the cylinder Effective bulk modulus Annulus area of cylinder chamber Half-volume of cylinder The i th load mass and load spring constant Viscous damping coefficient External load of the i th hydraulic actuator Desired and actual displacement of the i th cylinder The sign function The hyperbolic tangent function

observer (HGDOB) for one-degree-of-freedom plant may be not favorable to handle the coupled-disturbance with largely unknown dynamics. By referring to the HDOB in [42], this paper presented the CDOB in nonlinear backstepping controller design. Different from the HDOB in [41], this CDOB includes not only the independent element in diagonal formulas but also the coupled element in non-diagonal formulas as shown in Eq. (3). The comparison results with the flatness controller denote that the proposed approach can compensate for the coupled disturbances from the dynamic external loads more efficiently than the HDOB. Furthermore, the proposed controller also significantly improves the dynamic tracking performance of the Two-DOF robotic arm than the robust H∞ control approach.

2. Problem formulation In many EHSs, if the state vector is defined as

X = [x11, …, x14 , …, xi1, …, xi4 , …, x n1, …, x n4 ]T = [y1 , y1̇ , pL1, x v1, …, yi , yi̇ , pLi , x vi , …, yn , yṅ , pLn , x vn]T ,

(1)

then the dynamic model and the output equation of the multi-dof manipulator with n EHAs are [43]

⎡ xi̇ 1⎤ ⎢ ⎥ ⎢ xi̇ 2 ⎥ Ẋi = ⎢ ⎥ ẋ ⎢ i3 ⎥ ⎢⎣ xi̇ 4 ⎥⎦ ⎡ 0 1 0 ⎢ A K b p ⎢− − ⎢ m m m i i i ⎢ 4βe A p 4βe Ctl =⎢ − − ⎢ 0 Vt Vt ⎢ ⎢ 0 0 ⎢ 0 ⎣

⎤ ⎥ ⎥ 0 ⎥⎡ xi1⎤ ⎥⎢ xi2 ⎥ ⎥⎢ ⎥ ps − sgn(xi4 )xi3 ⎥⎢ xi3 ⎥ ⎥⎢⎣ xi4 ⎥⎦ ⎥ 1 − ⎥ Tsvi ⎦ 0

4βe CdwKsv Vt ρ

⎡ 0⎤ ⎡ 0 ⎤ ⎢ ⎥ ⎥ ⎢ 0 ⎢ FLi ⎥ ⎥ ⎢ + ⎢ 0 ⎥ui − ⎢ mi ⎥, ⎢ ⎥ ⎢K ⎥ ⎢ 0⎥ ⎢ svi ⎥ T ⎣ 0⎦ ⎣ svi ⎦ yi = xi1, i = 1, …, n.

(2)

Assumption 1. To illustrate the problem of coupled disturbance, the multi-dof manipulator is considered to be the Robotic Bigdog, which is fully-actuated by several EHAs as shown in Fig. 1. The

Please cite this article as: Guo Q, et al. Coupled-disturbance-observer-based position tracking control for a cascade electro-hydraulic system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.014i

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external load FL acts on the top of the manipulator, where FL may be constant or dynamic, but bounded by FLmax [13]. The motion control problem of the multi-dof manipulator is to design a cascade controller ui(i = 1, … , n), which guarantees not only all the states X stable but also the outputs yi (i = 1, … , n) tracking the demand inputs yid (i = 1, … , n) under the largely unknown load disturbances FLi/mi existed in the EHAs.

3. Modelling design of Two-DOF manipulator robotic arm 3.1. Coupled disturbance observer design

Remark 1. Each of the external loads FLi is uncorrelated. Meanwhile, FLi is affected by the position magnitude and motion frequency of all EHAs, i.e., FLi(t ) = FLi(t , FL, θ1, θ1̇ , θ¨1, …θi, θi̇ , θ¨i, …θn, θṅ , θ¨n), where θi, θi̇ , θ¨i are joint angle, angular velocity, angular acceleration. From Remark 1, if the unknown disturbances are defined as di = FLi/mi(i = 1, … , n), then the nth orders CDOB is designed as follow

⎡⎤ ⎡ ⎢ d1 ⎥ ⎢ −Kd11 ⎢ ⋮⎥ ⎢ ⋮ ⎢⎥ ⎢ ⎢ di ⎥ = ⎢ −Kdi1 ⎢ ⋮⎥ ⎢ ⋮ ⎢ ⎥ ⎢ −K ⎣ d ⎦ ⎣ dn1 n

⋯ −Kd1i ⋱ ⋮ ⋯ −Kdii ⋱ ⋮ ⋯ −Kdni

⎡x ⎢ 12 ⎢ ⎢ x i2 ⎢ ⎢ ⎢⎣ x n2 n×n

⋯ −Kd1n ⎤ ⎥ ⋱ ⋮ ⎥ ⋯ −Kdin ⎥ ⎥ ⋱ ⋮ ⎥ ⋯ −Kdnn ⎥⎦

− ⋮ − ⋮ −

x12 ⎤ ⎥ ⎥ xi2 ⎥ ⎥ ⎥ xn2 ⎥⎦

0⋯

0

0

0

0⋯

0 0 0⋯ Ap K b 0⋯ ⋯0 − − − mi mi mi 0 0 ⋯0 0 ⋱ ⋱

⋯0

0

0

0

0

⋯0

(3)

⎤ 0 0 0 ⎥ ⎥ ⎥ 0 0 0 ⎥ ⎥ 0 0 0 ⎥ ⎥ 0 0 0 ⎥ ⎥ Ap ⎥ K b − − − mn mn m n ⎥⎦ 4n × n

⎡  ⎤ ⎢ d1 ⎥ ⎢ ⋮⎥ ⎢ ⎥ X − ⎢ di ⎥ . ⎢ ⋮⎥ ⎢ ⎥ ⎣ dn ⎦

̇ From (3), the derivatives di(i = 1, … , n) are given by

)

(

)

(

)

(5)

After (2) and (4) are substituted into (5), the observer error equations are given by

d˜i̇ = − Kdi1d˜1 − … − Kdiid˜i… − Kdind˜ n + di̇ , where the observer errors d˜ i = di − di(i = 1, … , n).

Next the stability of the CDOB is analysed to guarantee the observer error d˜ i bounded and arbitrarily small by the observer gain regulation. Theorem 1. The disturbance derivative di̇ is assumed to be bounded ̇ for i ¼1,…,n, if there exists one Hurwitz matrix A such that by dimax

(7)

where Kdii > 0(i = 1, … , n) and A is diagonally dominant, then the disturbance observer errors d˜ i(i = 1, … , n) are bounded-input and bounded - output(BIBO), and can be arbitrarily small by increasing the positive constants Kdii(i = 1, … , n).

(4)

̇ ̇ − … − K ẋ − x̇ … − K ̇ ̇ − x12 ̇ di = − Kdi1 x12 dii i2 i2 din x n2 − x n2 .

i

motion of the j th EHA. Thus, the independent HDOB is a particular form of the CDOB as the coupled elements in (3) are considered to be zeros.

⎡ −Kd11 ⋯ −Kd1n ⎤ ⎥ ⎢ ⋮ ⋮ ⎥ , A=⎢ ⋮ ⎢⎣ −Kdn1 ⋯ −Kdnn ⎥⎦ n×n

Remark 2. For convenient illustration, the CDOB block diagram for Two-DOF robotic arm is constructed in Fig. 3. For CDOB 1, the disturbance estimation d1 is obtained by (3) using the state xij and the dynamic estimations xij for i = 1, 2, j = 2. Then the dynamic estimation x12 is obtained by (4) using the states xij for i = 1, j = 1, 2, 3 and d1. The constructed form of the CDOB 2 is similar to CDOB 1.

(

Definition 1. As shown in (3), (4), if Kdij ≠ 0(i ≠ j ), this disturbance observer is called as CDOB. This CDOB is different from general high-gain disturbance observer (HDOB) [42], since the coupled elements are designed in dynamic disturbance estimation. This proportional gain K denotes the weight for d caused by the dij

where di and xi2 are the dynamic estimations of di and xi2, and Kdij(i, j = 1, … , n) are the observer gains. The variables xi2(i = 1, … , n) are designed to estimate the states x i2(i = 1, … , n) as follow ⎡ K Ap b ⎢− − − m1 m1 ⎡ x̇ ⎤ ⎢ m1 ⎢ 12 ⎥ ⎢ 0 0 0 ⎢ ⋮ ⎥ ⎢ ⎢ ̇ ⎥ ⎢ =  0 0 0 x ⎢ ⎢ i2 ⎥ ⎢ ⋮ ⎥ ⎢ ⎢ 0 0 0 ⎢ ̇ ⎥ ⎣ xn2 ⎦ ⎢ ⎢ 0 0 0 ⎢⎣

Fig. 1. Multi-dof manipulator driven by several EHAs in lateral plane.

(6)

Proof. From Assumption 1 and Remark 1, the external load FL ∈ C1 and bounded by FLmax , then the external load on each EHA FLi ∈ C1. Thus, the disturbance derivative di̇ can be assumed to be bounded ̇ for i = 1, … , n. by dimax The observer error Eq. (6) are described as follow

⎡ ˜̇ ⎤ ⎡ ⎤⎡ ˜ ⎤ ⎡ ̇ ⎤ ⎢ d1 ⎥ ⎢ −Kd11 ⋯ −Kd1n ⎥⎢ d1 ⎥ ⎢ d1 ⎥ ⎢ ⋮⎥=⎢ ⋮ ⋮ ⋮ ⎥⎢ ⋮ ⎥ + ⎢ ⋮ ⎥, ⎢ ̇ ⎥ ⎢ −K ⎥⎢ ˜ ⎥ ⎢ ̇ ⎥ ⎣ d˜ n ⎦ ⎣ dn1 ⋯ −Kdnn ⎦⎣ dn ⎦ ⎣ dn ⎦

(8)

where di̇ and d˜ i are considered as system input and output for i¼1, …,n. The integration of (8) is

d˜(t ) = e At d˜(0) + If q(t ) =

∫0

t

e A(t − τ )d(̇ τ )dτ .

t

(9) 2

n ̇ = ∑i = 1 di̇ max , then the fol∫0 e A(t − τ )d(̇ τ )dτ , and dmax

lowing condition is satisfied [46]

Please cite this article as: Guo Q, et al. Coupled-disturbance-observer-based position tracking control for a cascade electro-hydraulic system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.014i

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qi(t ) ≤

⎛ ⎜ ⎝

( A e d(̇ t )) ⎞⎠ ⎛ ⎞ σ A ) + σ(A ) (e ) , ⎝ ( ⎠

(A

)

d(̇ t )

−1

̇ ⎜ ≤ dmax

i

−1 At

+

−1

i

zi1 = xi1 − yid zij = xij − αi(j − 1),

⎟

i

i

−1

At

(10)

i

( )

A  1. Since A is Hurwitz, a finite time T0 exists such that

(e )

ij

( ), −1

≤ σmin A

∀ t ≥ T0,

(11)

(i × (j − 1))th order virtual control. To conveniently derive the backstepping controllers, the extended state error vector zex and the related functions are defined as follow

zex = [z11, z12, z13, z14 , d˜1, …, zi1, zi2, zi3, zi4 , d˜i , …, T z , z , z , z , d˜ ] n1

n2

n3

for i, j = 1, … , n. Eq. (11) denotes that the n-dimensional column vector

fi1 = ( − Kxi1 − bxi2)/mi

(e At )i ≤ nσmin(A−1). Since 0 < σmin(A−1) ≤ σi(A−1), after (11) is sub-

fi2 = −

stituted into (10), the following inequality is obtained

fi3 = − xi4 /Tsvi

( )( 1 + nσ ( A )) , ( σ ( A ) + nσ ( A ))

̇ σ A−1 qi(t ) ≤ dmax i ̇ ≤dmax

i

−1

2 i

−1

∀ t ≥ T0. (12)

2 n If the initial disturbance error is defined as |d˜ (0)|= ∑i = 1 d˜ i (0) ,

( )( n d˜(0)

Vt

x i2 −

4βe Ctl Vt

( ))

̇ ̇ σ A−1 , + dmax + ndmax i

gi2 =

x i3 ,

4βe CdwKsv Vt ρ

(15)

n

i = 1, …, n.

ps − tanh(kxi4 )xi3

gi3 = Ksvi/Tsvi

(16)

Theorem 2. Consider the cascade EHS (2) and the CDOB (3), (4). If the following three conditions such that

then from (9), the following condition is satisfied

d˜i(t ) ≤ σi A−1

4βe A p

n4

gi1 = A p /mi

−1

min

(14)

where yid is the demand position of the i th cylinder, αi(j − 1) is the

⎟

where σi A−1 (i = 1, … , n) are singular values of the inverse matrix

At

i = 1, …, n, j = 2, 3, 4

(13)

for ∀ t ≥ T0 , i¼ 1,…,n. Since the diagonal gains Kdii(i = 1, … , n) are designed as highgain positive constants, and the matrix A is diagonally dominant, −1

then the singular values σi(A ) can be reduced by increasing Kdii for i¼1,…,n. Thus, from (13), the observer error d˜ i is BIBO, especially can be arbitrarily small by increasing these high-gains. □

(I) the cascade backstepping controller [44]

αi1 = − ki1zi1 + yiḋ αi 2 =

− fi1 − zi1 + di + αi̇ 1 /gi1 , − ki3zi3 − gi1zi2 − fi2 + αi̇ 2 /gi2

( −k z

)

i2 i2

( ) ui = ( − ki4zi4 − gi2zi3 − fi3 + αi̇ 3) /gi3 αi 3 =

(17)

where k ij(i = 1, … , n, j = 1, … , 4) are positive constants, 3.2. Cascade controller design In this section, the backstepping controllers are designed for this cascade EHS based on the CDOB. The coupled elements in the CDOB will be handled by Lyapunov technique to guarantee the position tracking error ultimate boundedness. The i th general backstepping iteration is shown as Fig. 2. Remark 3. As most functions in the design of backstepping controller are required to be smooth, the sign function sgn(xi4) in (2) is replaced by the hyperbolic tangent function tanh( kx i4 ), where k is a positive constant [45]. The system state errors zij(i, j = 1, … , n) are defined as

1

(II) the diagonal gains Kdii > 4k for i¼1,…,n, i2 (III) the non-diagonal gains Kdij = − Kdji(i ≠ j ) for i, j = 1, … , n, are all satisfied, then zex(t) is ultimate boundedness [47] and its convergence domain is an hypersphere Hr,

⎧ ⎛ 1 ˜ 2 ⎞⎫ 2 ⎪ di ) ⎟ ⎪ ⎜ ki1zi1 + ki2(zi2 + 2k i 2 ⎪ ⎜ ⎟⎪ n ⎪ ⎜ ⎟⎪ 2 2 + + k z k z ⎪ zex(t )| ∑ ⎜ i3 i3 i4 i4 ⎟⎪ ⎪ 2 i=1 ⎜ ⎟⎪ ⎪ ⎪ ⎛ ⎞ Hr = ⎨ ⎜ +λ ⎜ d˜ − 1 d ̇ ⎟ ⎟⎬ , i i i ⎜ ⎟ ⎪ ⎪ 2λ i ⎠ ⎝ ⎝ ⎠⎪ ⎪ ⎪ ⎪ ̇2 ⎪ n di ⎪ max = ⎪ ∑ ⎪ ⎪ ⎪ 4 λ i ⎩ i=1 ⎭ where ∀ t > T1 (T1 is a finite time), and λ i = Kdii −

(18) 1 (i 4k i2

= 1, … , n).

Proof. Step 1: The derivative z11 is given by

̇ = z12 + α11 − y1̇ d . z11

(19)

The Lyapunov candidate function V11 is defined as

V11 =

1 2 z11. 2

(20)

If (14) is substituted into (20), then the derivative of V11 is obtained by

̇ = z (z + α − ẏ ). V11 11 12 11 1d Fig. 2. The i

th

general backstepping iteration design.

After the virtual control

(21)

α11 in (17) is substituted into (21), the

Please cite this article as: Guo Q, et al. Coupled-disturbance-observer-based position tracking control for a cascade electro-hydraulic system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.014i

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Fig. 3. The block diagram of the control system for Two-DOF robotic arm.

derivative of V11 is given by

̇ = − k z 2 − k z 2 − z d˜ + g z z V12 11 11 12 12 12 1 11 12 13 n

̇ = − k z2 + z z , V11 11 11 11 12

(22)

− d˜1 ∑ Kd1id˜i + d˜1d1̇ i=1

where z11z12 is a cross-term removed by the subsequent backstepping iteration. From (14) and (16), the derivative of z12 is given by

⎛ 2 ⎞ 1 2 z12d˜1⎟ + g11z12z13 ≤ − k11z11 − k12⎜ z12 + k12 ⎝ ⎠ n

2 − Kd11d˜1 − d˜1 ∑ Kd1id˜i + d˜1 d1̇ i=2

̇ = x12 ̇ − α11 z12 ̇ = f11 + g11x13 − d1 − α11 ̇ .

(23)

If the Lyapunov candidate function V12 is defined as

1 2 1 2 V12 = V11 + z12 + d˜1 , 2 2

(24)

̇ is and (6), (14), (23) are substituted into (24), the derivative V12 given by

2 ⎛ 1 ˜⎞ 2 d1⎟ + g11z12z13 ≤ − k11z11 − k12⎜ z12 + 2k12 ⎠ ⎝ 2 2 n d1̇ ⎛ 1 ̇ ⎞ max d1 ⎟ + − λ1⎜ d˜1 − − d˜1 ∑ Kd1id˜i . 2λ 1 ⎠ 4λ 1 ⎝ i=2

Similarly, the derivative of z13 is given by

̇ = x13 ̇ − α12 z13 ̇ = f12 + g12x14 − α12 ̇ . After

̇ = V̇ + z z ̇ + d˜ d˜ ̇ V12 11 12 12 1 1 n

(28)

If the Lyapunov candidate function V13 is defined as

− d1 − α11 ̇ ) − d˜1 ∑ Kd1id˜i + d˜1d1̇ .

(25)

i=1

V13 = V12 + If

(27)

̇ is given by α13 in (17) is substituted into (27), z13

̇ = − k13z13 − g11z12 + g12z14 . z13

2 = − k11z11 + z12(z11 + f11 + g11(z13 + α12)

(26)

̇ is given by α12 in (17) is substituted into (25), V12

1 2 z13, 2

(29)

̇ is and (26) and (28) are substituted into (29), the derivative V13

Fig. 4. The simulation results of the robust H∞ controller in case 1.

Please cite this article as: Guo Q, et al. Coupled-disturbance-observer-based position tracking control for a cascade electro-hydraulic system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.014i

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Fig. 5. The simulation results of the flatness controller with PIDOB in case 1.

given by

Fig. 6. The simulation results of the proposed controller with CDOB in case 1.

̇ = x14 ̇ − α13 z14 ̇ = f13 + g13u1 − α13 ̇ . 2

⎛ ⎞ ̇ ≤ − k z 2 − k ⎜ z + 1 d˜ ⎟ − k z 2 + g z z V13 12 12 1 13 13 11 11 12 13 14 2k12 ⎠ ⎝ 2

⎛ 1 ̇ ⎞ d1 ⎟ + − λ1⎜ d˜1 − 2λ 1 ⎠ ⎝

d1̇

2 max

4λ 1

After

− d˜1 ∑ Kd1id˜i . i=2

Then, the derivative of z14 is given by

̇ is given by α14 in (17) is substituted into (31), z14

̇ = − k14z14 − g12z13. z14

n

(31)

(32)

If the Lyapunov candidate function V14 is defined as

(30) V14 = V13 +

1 2 z14 , 2

(33)

Please cite this article as: Guo Q, et al. Coupled-disturbance-observer-based position tracking control for a cascade electro-hydraulic system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.014i

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Fig. 7. The simulation results of the robust H∞ controller in case 2.

̇ is after (30) and (32) are substituted into (33), the derivative V14 given by

Step n: Similar to the derivation in Step i, the Lyapunov candidate function Vn1 is

⎛ ⎞2 ̇ ≤ − k z 2 − k ⎜ z + 1 d˜ ⎟ − k z 2 − k z 2 V14 11 11 12 12 1 13 13 14 14 2k12 ⎠ ⎝

Vn1 = V(n − 1)4 +

2 2 n d1̇ ⎛ 1 ̇ ⎞ max ˜ d1 ⎟ + − λ1⎜ d1 − − d˜1 ∑ Kd1id˜i . 2λ 1 ⎠ 4λ 1 ⎝ i=2

1 2 zi1, 2

(34)

(35)

̇ , V̇ is given by then similar to the derivation of V11 i1

Vi̇ 1 = V(̇ i − 1)4 − ki1zi21 + zi1zi2,

(40)

̇ , V̇ , V̇ and V̇ are given by and the derivatives Vn1 n2 n3 n4

Step i (2 ≤ i ≤ n − 1): If the Lyapunov candidate function Vi1 is defined as

Vi1 = V(i − 1)4 +

1 2 z n1, 2

Vṅ 1 = V(̇ n − 1)4 − k n1z n21 + z n1z n2,

(41)

2 ⎛ 1 ˜ ⎞ Vṅ 2 ≤ V(̇ n − 1)4 − k n1z n21 − k n2⎜ z n2 + dn⎟ 2k n2 ⎠ ⎝ 2 2 dṅ ⎛ 1 ̇ ⎞ max ˜ + gn1z n2z n3 − λ n⎜ dn − dn ⎟ + 2λ n 4λ n ⎝ ⎠ n− 1

(36)

− d˜ n ∑ Kd1jd˜j ,

(42)

j=1

for i ≥ 2. And so on, Vi2̇ , Vi3̇ and Vi4̇ are given by

2 ⎛ 1 ˜ ⎞ Vṅ 3 ≤ V(̇ n − 1)4 − k n1z n21 − k n2⎜ z n2 + dn⎟ − k n3z n23 2k n2 ⎠ ⎝

2

⎛ 1 ˜⎞ Vi̇ 2 ≤ V(̇ i − 1)3 − ki1zi21 − ki2⎜ zi2 + di⎟ + gi1zi2zi3 2k i 2 ⎠ ⎝

2 2 dṅ ⎛ 1 ̇ ⎞ max ˜ dn ⎟ + − λ n⎜ dn − 2λ n 4λ n ⎝ ⎠

2

2 di̇ ⎛ 1 ̇ ⎞ max di ⎟ + − λ i⎜ d˜i − 2λ i ⎠ 4λ i ⎝

n− 1

− d˜ n ∑ Kd1jd˜j + gn2z n3z n4 .

n

− d˜i

∑

Kd1jd˜j ,

2 ⎛ 1 ˜ ⎞ Vṅ 4 ≤ V(̇ n − 1)4 − k n1z n21 − k n2⎜ z n2 + dn⎟ − k n3z n23 2k n2 ⎠ ⎝

2

⎛ 1 ˜⎞ Vi̇ 3 ≤ V(̇ i − 1)4 − ki1zi21 − ki2⎜ zi2 + di⎟ − ki3zi23 2k i 2 ⎠ ⎝

2 2 dṅ ⎛ 1 ̇ ⎞ max dn ⎟ + − k n4z n24 − λ n⎜ d˜ n − 2λ n 4λ n ⎝ ⎠

2

2 di̇ ⎛ 1 ̇ ⎞ max di ⎟ + − λ i⎜ d˜i − 2λ i ⎠ 4λ i ⎝

n− 1

− d˜ n ∑ Kd1jd˜j .

n

− d˜i

∑

Kd1jd˜j + gi2zi3zi4 .

j = 1, j ≠ i

−

2 di̇ ⎛ ⎞2 1 max di̇ ⎟ + − λ i⎜ d˜i − 2λ i ⎠ 4λ i ⎝

∑ j = 1, j ≠ i

Kd1jd˜j .

Since Kdij = − Kdji(i ≠ j ) for i, j = 1, … , n, if Vi4̇ (i = 1, … , n − 1) ̇ is given by are substituted into (44), Vn4

Vṅ 4 ≤ −

n

⎛

⎛

i=1

⎝

⎝

∑ ⎜⎜ ki1zi21 + ki2⎜ zi2 +

2 ⎞ 1 ˜⎞ di⎟ + ki3zi23 + ki4zi24⎟⎟ 2k i 2 ⎠ ⎠

2 ⎛ 1 ̇ ⎞ − ∑ λ i⎜ d˜i − di ⎟ + 2λ i ⎠ i=1 ⎝ n

n

− d˜i

(44)

j=1

(38)

2 ⎛ 1 ˜⎞ Vi̇ 4 ≤ V(̇ i − 1)4 − ki1zi21 − ki2⎜ zi2 + di⎟ − ki3zi23 2k i 2 ⎠ ⎝

ki4zi24

(43)

j=1

(37)

j = 1, j ≠ i

n

∑ i=1

di̇

2 max

4λ i

(45)

(39)

Please cite this article as: Guo Q, et al. Coupled-disturbance-observer-based position tracking control for a cascade electro-hydraulic system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.014i

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Fig. 8. The simulation results of the flatness controller with PIDOB in case 2.

Since Kdii >

1 , 4k i2

then λ i > 0 for i¼1,…,n. All elements are ne-

gative except the last one in (45). Thus, the extended state error vector z (t ) is ultimate boundedness. Let V̇ = 0, the convergence ex

n4

domain Hr of zex is obtained as shown in (18).

□

Remark 4. In the cascade backstepping controller (17), several virtual control derivatives α̇ij, i = 1, … , n, j = 1, 2, 3 are needed to derive the final controller u. To solve this problem, some techniques such as difference approximation [20], state analytic method [27], and dynamic surface control (DSC) method [39] are proposed.

Fig. 9. The simulation results of the proposed controller with CDOB in case 2.

The difference approximation of derivative may cause the “explosion of complexity” due to repeated derivations of virtual control inputs [31]. The state analytic method is also complicated resulted from the multiple model functions. However, the DSC method not only prevents the complexity problem, but also relaxes the smoothness requirement on plant models and desired signals [32]. Thus, in the future, the virtual control derivatives should be handled by the DSC. For convenient illustration, the block diagram of the control system for Two-DOF robotic arm is shown Fig. 3. The external load

Please cite this article as: Guo Q, et al. Coupled-disturbance-observer-based position tracking control for a cascade electro-hydraulic system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.014i

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Table 1 Hydraulic parameters used in simulation and experiment. Parameter

Value

Parameter

Value

Cd ps Vt

0.62 40 bar

w Ap βe

0.024 m 4.91 cm2 7  108 Pa

L max umax b ρ

58 mm

8.74 × 10−5 m3 5  10  4 m/V

Ksv

10 ms 0

Tsv K Ctl

2.5 × 10−11 m3/(s·Pa)

10 V 2200 Ns/m 850 kg/m3

Table 2 Mechanical parameters used in simulation.

Fig. 10. The experimental bench.

disturbance di(i = 1, 2) on each EHA is estimated by CDOB (3), (4) using the state xij for i = 1, 2, j = 1, 2, 3. The control input of each EHA ui(i = 1, 2) is designed by backstepping iteration (17) using x (i = 1, 2, j = 1, … , 4) and the disturbance estimation d (i = 1, 2).

Parameter

Value

Parameter

Value

m1 mf

3.347 kg 1.068 kg

m2 I1

1.498 kg

I2f

0.045 kg·m2 0.16 m

I2

2

P1P2

0.030 kg·m 0.35 m

P2Pm2

0.12 m

0.134 kg·m2

P1Pm1 εm1

7.9°

i

ij

αi1 = − ki1zi1 + yiḋ − fi1 − zi1 + di + αi̇ 1 /gi1 , − ki3zi3 − gi1zi2 − fi2 + αi̇ 2 /gi2

4. Simulations

αi 2 =

( −k z

To verify the proposed controller, a Two-DOF robotic arm is considered to be the typical multi-dof manipulator driven by two cascade EHAs. This robotic arm is from one leg of Robotic Bigdog, which is manufactured based on the HyQ-Robot [48] of Italian Institute of Technology as shown in Fig. 10.

αi 3 =

(

4.1. Parameters design of robotic arm From (3) and (4), the 2th orders CDOB is given by

d1 = − Kd11 x12 − x12 − Kd12 x22 − x22

(

)

(

),

d2 = − Kd21(x12 − x12) − Kd22(x22 − x22) ̇ = x12 ̇ x22

1 ( − Kx11 − bx12 + A p x13) − d1 m1f

1 = ( − Kx21 − bx22 + A p x23) − d2 m2f

(46)

. (47)

Similarly from (17), the cascade backstepping controller is given by

)

i2 i2

)

ui = ( − ki4zi4 − gi2zi3 − fi3 + αi̇ 3)/gi3

i = 1, 2. (48)

In practice, since there exist unknown parametric uncertainties in some hydraulic parameters such as the effective bulk modulus βe, the viscous damping coefficient b, the coefficient of the total leakage of the cylinder Ctl, and the density of hydraulic oil ρ, the elements fij, gij in (48) cannot be computed easily. In this study, only the nominal parameters of this EHS are just considered both in simulation and experiment, which are obtained by experimental parameter identification and measurement as shown in Table 1. In simulation, the external loads FL1, FL2 on two EHAs are computed by Lagrange equation using some mechanical parameters of the robotic arm, which are shown in Table 2. Notation 1. L max is the stroke of hydraulic cylinder, m1 is the upper arm mass, m2 is the forearm mass, mf is the disc mass, I1 is the inertia of the upper arm around the shoulder, I2 is the inertia of the forearm around the elbow, I2f is the inertia of the forearm around the elbow including load, P1P2 is the length of the upper arm, P1Pm1 is the distance from the upper arm centroid to shoulder,

Fig. 11. The experimental results of robust H∞ controller in case 1.

Please cite this article as: Guo Q, et al. Coupled-disturbance-observer-based position tracking control for a cascade electro-hydraulic system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.014i

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Fig. 12. The experimental results of the flatness controller with PIDOB in case 1.

P2Pm2 is the distance from the forearm centroid to elbow, and εm1 is the eccentric angle of cylinder to shoulder. 4.2. Simulation results discussion

Fig. 13. The experimental results of the proposed controller with CDOB in case 1.

of the proposed controller, the motion frequencies of two EHAs are not greater than 1 Hz. Otherwise, the control saturation is unavoidable. (I) Comparison with robust H∞ controller

If the control parameters are designed as k11 = k21 = 5000, k12 = k22 = 500 , k13 = k23 = 50, k14 = k24 = 10, Kd11 = 100 , Kd12 = 10, Kd21 = − 10, Kd22 = 100, then some comparison results with the other two advanced controllers are demonstrated to verify the performance of the proposed method. Considering the bandwidth

The robust H∞ controller of EHS is proposed in [12] to address some parametric uncertainties such as the flow gain of the valve Kq, the viscous damping coefficient b, etc. This controller is obtained by solving the semidefinite programming problem (SDP):

Please cite this article as: Guo Q, et al. Coupled-disturbance-observer-based position tracking control for a cascade electro-hydraulic system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.014i

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min

γ

subjectto: ⎡ ¯ ¯T ⎢ AP + PA ⎢ ⎢ ⎢⎣ C¯1P

P > 0,

γ∈ R, P ∈ Rn × n, F ∈ Rm × n

xi*2 = yiḋ − k1v(xi1 − yid ) xi*3 =

T + B¯2F + FT B¯2 T B¯1

+ D¯ 12F

⎤ ⋆ ⎥ ⎥ −γI ⋆ ⎥ < 0, D¯ 11 −γI ⎥⎦ ⋆

+ (b − mif k1v − mif k2)ei2 + di ] (49)

where A¯ , B¯1, B¯2, C¯1, D¯ 11, D¯ 12 are the augmented matrices of the extended dynamic model of EHS, ‘⋆’ is the transpose of the offdiagonal part, and the matrix J is the Jordan form. If an optimal solution (γ, P, F) is obtained from the SDP (49) by linear matrix inequality method, then the state feedback controller can be given by u = FP −1(yd − y ). This robust controller is usually high orders when the iterative optimization terminates. Then the controller can be simplified to fourth order by Balanced Truncation Method [49] as follow

n4 s 4 + n3s 3 + n2s 2 + n1s + n0 d4s 4 + d3s 3 + d2s 2 + d1s + d0

( yid − yi ), i = 1, 2

(50)

where n4 ¼0.032, n3 ¼1.0, n2 ¼112.4, n1 ¼3145.2, n0 ¼3.5, d4 ¼1, d3 ¼53.7, d2 ¼823.6, d1 ¼2427.5, d0 ¼37.8. (II) Comparison with flatness controller with DOB The flatness controller of EHS is proposed in [41] to compensate for the external load disturbance FL by proportion-integral-disturbance-observer (PIDOB), which is described as

di = − k p(xi2 − xi2) − ki xi̇ 2 =

∫0

t

(xi2 − xi2)dt

1 ( − Kxi1 − bxi2 + A p xi3) − di mif

,

i = 1, 2 (51)

where kp and ki are PI gain of the disturbance observer. If the new tracking errors eij(i = 1, 2; j = 0, 1, 2, 3) are defined as

ei0 =

∫0

t

(xi1 − yid )dt

ei1 = xi1 − yid

1 [mif y¨id + byiḋ + Kyid − mif k 0ei0 Ap + (K − bk1 + mif k12v − mif k1)ei1

¯ + B¯ F − PJ = 0, AP 2

ui (s ) =

11

,

ei2 = xi2 − xi*2 ei3 = xi3 − xi*3

(52)

then the virtual control variables x i*2, x i*3, x i*4 are given by

xi*4 =

γ1xi2 + γ2xi3 − k3ei3 + xi̇ *3 γ3i ps − sgn(xi4 )xi3

(53)

where the constant parameters γ2 = 4βeCtl/Vt , γ1 = 4βeAp /Vt , γ3i = 4βeCdwKsvi/Vt ρ , the control parameters kp ¼ 30, ki ¼300, k0 = 100, k1 = 5000, k2 = 500, k3 = 50, k1v = 100. Thus, the flatness controller can be given by

ui = (Tsvixi̇ *4 + xi*4 )/Ksvi, i = 1, 2.

(54)

To illustrate the problem, two cases of the cylinder position demands are considered as. (I) y1d = 0.5L maxsin(0.5πt ), y2d = 0.5L maxsin(πt );. (II) y1d = 0.5L maxsin(1.6πt ), y2d = 0.5L maxsin(2πt ). From Figs. 4 to 6 show the cylinder position responses of the three controllers in case 1, where the motion frequencies of two EHAs are 0.25 Hz, 0.5 Hz respectively. Since the external loads on two EHAs are well estimated by the disturbance observer, the proposed controller (i.e., Backstepping controller with CDOB) has high position tracking accuracy 0.1 mm than that of the robust H∞ controller 1 mm, as shown in Fig. 4a and Fig. 6a. However, at the cost of the control variable as shown in Fig. 4b and Fig. 6c, the position tracking error can be further reduced by the proposed controller. Fig. 5b and Fig. 6b show that the disturbance estimation of the PIDOB is relative poorer than that of the proposed CDOB. This phenomenon denotes that the external load on each of EHA is related with the individual's motion. If the disturbance is independently estimated by the PIDOB (i.e., Kd12 = Kd21 = 0), the estimation error is obvious bias as shown in time slices (2.5, 3.5)s and (6.5, 7.5)s of Fig. 5b. Then the demand frequencies of two EHAs are increased to 0.8 Hz, 1 Hz, respectively. That is a critical condition where every external load approaches the maximal load capability respectively. The position tracking error of the robust H∞ controller without disturbance compensation is increased to 5 mm as shown in Fig. 7a, especially asymmetric bias in time slices (2, 3)s, (6, 7)s, which is caused by the coupled disturbance effect. Thus, the disturbance observer is designed to estimate both the independent element and

Fig. 14. The experimental results of robust H∞ controller in case 2.

Please cite this article as: Guo Q, et al. Coupled-disturbance-observer-based position tracking control for a cascade electro-hydraulic system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.014i

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the coupled element as shown in Fig. 8b and Fig. 9b. After the disturbances are well compensated by the proposed controller, the dynamic error is declined and the asymmetric bias eliminates as shown in Fig. 8a and Fig. 9a. In case 2, the position tracking error of the proposed controller is still smaller than the other two controller as shown in Fig. 7a, Fig. 8a and Fig. 9a. However, the control voltage of the proposed controller is the largest one among three controllers, which approach the saturation umax = 10 V as shown in Fig. 7c, Fig. 8c and Fig. 9c. If the motion frequency of this robotic arm is further increased, the proposed controller cannot guarantee the stability of this cascade EHS.

5. Experimental results 5.1. Experiment set up To verify the proposed controller, the experimental bench of the Two-DOF robotic arm driven by EHA is set up as shown in Fig. 10. The two electro-hydraulic actuators includes two servo valves (FF102/03021T240), two single-rod cylinders (UG1511R25/16-80), a pump station (HY-36CC-01/11kw), and two accumulators (NXQ1L1.0/31.5H). The cylinder position is measured by displacement transducer (JHQ-GA-40). The control algorithm is executed by industrial personal computer (IPC). The load mass is a disc on the forearm terminal. Some hydraulic and mechanical parameters of this experimental bench are shown in Tables 1 and 2. 5.2. Experimental results discussion Figs. 11–16 show the corresponding experimental results of the two working conditions mentioned in the simulation section. When the motion frequencies of two EHAs are low, the external load on the shoulder EHA is not greater than 200 m/s2. Although the motion frequencies of elbow is higher, its external load is relative smaller than that of shoulder as shown in Fig. 12b and Fig. 13b. The position tracking error of the proposed controller with CDOB is less than 1 mm, better than that of the other two controllers as shown in Fig. 11a, Fig. 12a and Fig. 13a. Furthermore, the dynamic error with asymmetric bias is obvious existed in the result of the robust H∞ controller but eliminated in that of the other two controllers with disturbance observer. This indicates that the external load is well compensated by the disturbance observer. Due to the coupled disturbance existed in each EHA, the disturbance estimation of the PIDOB without coupled elements is relative poorer than that of the CDOB as shown in Fig. 11b and Fig. 12b, which is consistent with the simulation results as shown in Fig. 5b and Fig. 6b. The load pressures of the two EHAs are not greater than 14 bar as shown in Fig. 11b, especially less than the supply pressure 40 bar, which means that the dynamic behavior of this cascade EHS can be guaranteed by each controller. The control voltage requirement of the proposed controller is approximately 6 V, the largest one among three controllers, but not greater than the saturation 10 V as shown in Fig. 11b, Fig. 12c, and Fig. 13c. Then the motion frequency of upper arm is increased from 0.25 Hz to 0.8 Hz and the motion frequency of forearm is increased from 0.5 Hz to 1 Hz. The dynamic tracking errors of the robust H∞ controller without disturbance compensation are amplified to 3 mm along with the increased frequencies. Moreover, the low-frequency flutter of the position response exists in this controller as shown in Fig. 14a. Simultaneously, the external loads on two EHAs approach 700 m/s2, 300 m/s2 as shown in Fig. 15b and Fig. 16b respectively. Thus, the load disturbance becomes the main element of the virtual control αi2 in (17), which would be compensated by the controller. On the other

Fig. 15. The experimental results of the flatness controller with PIDOB in case 2.

hand, the increased derivative of yiḋ will directly lead to the large value yiḋ of the virtual control αi1. Therefore, both of two latter controllers are more easier to approach the saturation than the robust H∞ controller as shown in Fig. 14b, Fig. 15c and Fig. 16c. Due to the large disturbance compensation in the disturbance observer, the good dynamic behavior without flutter and asymmetric bias is guaranteed. The position tracking error is reduced into 1 mm by the proposed controller with CDOB, which is the best one among three controllers as shown in Fig. 14a, Fig. 15a and Fig. 16a.

Please cite this article as: Guo Q, et al. Coupled-disturbance-observer-based position tracking control for a cascade electro-hydraulic system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.014i

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observer estimation error and the system state error convergence to a prescribed steady state level. The stability of the state feedback control system is proved by Lyapunov technique. The comparison results with the other two controllers have verified the effectiveness of the proposed controller, which is demonstrated in the motion control of Two-DOF robotic arm. In the future, there exist two typical problems to be addressed in cascade electro-hydraulic system. Firstly, some hydraulic parametric uncertainties caused by unknown viscous damping, physical characteristics of valve, bulk modulus and oil temperature variations will be handled by adaptive parametric law. Due to the ignorance of the negative effect by these uncertainties in this study, the controller is conservatively designed to guarantee the stability and performance of the EHS. On the other hand, the control capability of servo valve is limited in this study. This proposed approach may cause the control saturation since the backstepping iteration has produced several virtual control variables. Thus, the control saturation will be considered in electro-hydraulic system using dynamic surface control and anti-windup control method.

Acknowledgement This work was supported by the National Natural Science Foundation of China, China (No. 61305092, No. 51205045 and No. 61522302), and the Open Foundation of the State Key Laboratory of Fluid Power Transmission and Control (No. GZKF-201515).

References

Fig. 16. The experimental results of the proposed controller with CDOB in case 2.

6. Conclusion In this study, a state feedback controller embedded with a coupled disturbance observer is proposed for a cascade electrohydraulic system. This observer gain is designed as a diagonally dominant matrix form with coupled elements, which can estimate both the independent and the coupled disturbance from the external loads on more than two electro-hydraulic actuators. Simultaneously, the backstepping controller can compensate for the largely unknown disturbance estimation and guarantee both the

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Please cite this article as: Guo Q, et al. Coupled-disturbance-observer-based position tracking control for a cascade electro-hydraulic system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.014i