Extended-state-observer-based output feedback adaptive control of hydraulic system with continuous friction compensation

Available online at www.sciencedirect.com

Journal of the Franklin Institute 356 (2019) 8414–8437 www.elsevier.com/locate/jfranklin

Extended-state-observer-based output feedback adaptive control of hydraulic system with continuous friction compensation Chengyang Luo a, Jianyong Yao a,∗, Jason Gu b a School

of Mechanical Engineering, Nanjing University of Science & Technology, Nanjing 210094, PR China of Electrical & Computer Engineering, Dalhousie University, Halifax, NS B3J2X4, Canada

b Department

Received 16 February 2019; received in revised form 26 June 2019; accepted 9 August 2019 Available online 17 August 2019

Abstract This paper presents an extended state observer-based output feedback adaptive controller with a continuous LuGre friction compensation for a hydraulic servo control system. A continuous approximation of the LuGre friction model is employed, which preserves the main physical characteristics of the original model without increasing the complexity of the system stability analysis. By this way, continuous friction compensation is used to eliminate the majority of nonlinear dynamics in hydraulic servo system. Besides, with the development of a new parameter adaption law, the problems of parametric uncertainties are overcome so that more accurate friction compensation is realized. For another, the developed adaption law is driven by tracking errors and observation errors simultaneously. Thus, the burden of extended state observer to solve the remaining uncertainties is alleviated greatly and high gain feedback is avoided, which means better tracking performance and robustness are achieved. The designed controller handles not only matched uncertainties but also unmatched dynamics with requiring little system information, more importantly, it is based on output feedback method, in other words, the synthesized controller only relies on input signal and position output signal of the system, which greatly reduces the effects caused by signal pollution, measurement noise and other unexpected dynamics. Lyapunov-based analysis has proved this strategy presents a prescribed tracking transient performance and final tracking accuracy while obtaining asymptotic tracking performance in the presence of parametric uncertainties only. Finally, comparative experiments are conducted on a hydraulic servo platform to verify the high tracking performance of the proposed control strategy. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

∗

Corresponding author. E-mail address: [email protected] (J. Yao).

https://doi.org/10.1016/j.jfranklin.2019.08.015 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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1. Introduction Hydraulic systems can provide large and high tracking performance force/torque outputs with small size-to-power ratios, hence they attract considerable attention and are now playing indispensable roles in various of industrial applications such as mechanical auxiliary devices [1], vehicle technology [2], aircraft servo actuators [3], heavy manipulators [4], testing setup [5], and so on. However, because of the inherent nonlinear characteristics of hydraulic systems, modeling uncertainties and nonlinear dynamics are the main obstacles of developing highperformance controller [6,7]. As a result, a lot of works have been done to research and solve these problems [6–9]. Among the nonlinear factors which effect the performance of hydraulic systems, friction is the most complicated and difficult behavior to be solved. Since the mechanical contact phenomenon is hardly to be known, it is impossible to obtain the exact friction details to improve tracking performance [10,11]. Hence, conventional control schemes have to utilize simplified continuous linear friction model to overcome this problem [6]. However, for the increasing high-performance requirements such as low velocity servo control and precise motion control, Stribeck effects [12] and friction internal dynamics [13,14] can deteriorate the tracking performance significantly and even lead to instability in some cases. Model-based compensation has been proved to be a useful way to solve friction problems and has been employed widely in many applications [10,12,14,15]. Among the compensation methods, LuGre model [16] is one of the most common and effective tools utilized in those methods. It is because LuGre model reflects the average behavior of the contact surfaces in a microscopic view and presents it with a nonlinear first-order differential equation, which means LuGre model can capture the main frictional behaviors throughout a much simpler equation. However, there are still many practical difficulties so that LuGre model cannot be applied directly in some cases because its mathematical equations contain discontinuous terms. In practice, the discontinuous terms are the most obstacles since it is necessary for controller design to use the time derivation of LuGre friction when solving the unmatched nonlinear friction via backstepping [15,17]. In recent works, some solutions were developed to address this problem. [18] proposed a modified LuGre friction model by combing LuGre model and static friction model throughout a continuous transition. [19] identified the steady-state friction force by experiments firstly, and then rebuilt the static Stribeck function in LuGre model and replaced the signum function by the approximate value of static friction force. However, although the simple substitution can overcome the problem of time derivative in mathematics, it makes LuGre model unable to guarantee the properties of passivity and boundness. In [20], a novel continuous approximation of LuGre friction model was synthesized and applied in a nonlinear pneumatic servo system [21]. This new model successfully retains the features with regard to physical consistency and practicability of the initial model, which means it does not increase the complexity of the system stability analysis. However, even though the most accurate friction model cannot describe all the details exactly, there are always some unmodeled nonlinear parts in the system. Furthermore, the parameters in the friction model not only are hard to be determined but also may vary with the changes of experimental conditions. As a result, perfect compensation is impossible to realize. In addition, the aforementioned inherent nonlinear characteristics of hydraulic systems are also supposed to be taken into consideration to achieve high performance tracking control. In order to cope with these difficulties, many advanced nonlinear control strategies have been developed. For the nonlinear systems whose system parameters cannot be precisely known

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in advance, nonlinear adaptive control is a powerful tool to overcome parametric uncertainties and has been implemented widely [8,13,18,22,23,24], nevertheless, it is less helpful for solving uncertain nonlinearities. Sliding model control (SMC) [25] can solve the bounded modelling uncertainties and achieve excellent tracking performance but its discontinuous part needs to be replaced by continuous approximations to eliminate chatter or dither. [26] proposed a particular SMC method, namely adaptive robust integral of the sign of the error (ARISE), it obtains a continuous control input and an asymptotic tracking result. However, ARISE requires the high order derivative of command to be as smooth as possible, which is not applicable in many cases. Tseng et al. [27] employed fuzzy model to handle nonlinearities, as a result, the fuzzy tracking control design problem is parameterized in terms of a linear matrix inequality problem, which can be solved efficiently by convex optimization techniques. Besides, piecewise-model-based control [28] and Lipschitz-based methods [29] are also effective to solve complicated nonlinearities. In [30], a new control strategy named active disturbance rejection control (ADRC) was proposed to do with the unmodeled uncertainties and extern disturbances. In a creative way, ADRC estimates the generalized disturbances of considered system by employing an extended state observer (ESO) and then they can be handled by feed-forward compensation. Besides, ESO-based controller requires little model details while provides the states estimating information, which is considerably practical and has been verified in plenty of applications [31,32]. In [32], a linear ESO was developed, which can maximize the bandwidth to present practical optimization. However, despite the advantage of overcoming modeling uncertainties, ESO-based controller cannot cope with heavy parametric uncertainties, which would result in the deterioration of tracking performance and even instability. Hence, [33–35] merged adaptive control and ESO in one controller so that adaptive control can solve parametric uncertainties while ESO can solve uncertain nonlinearities. It is indeed an intelligent idea but [33] is unable to handle the unmatched nonlinearities. Besides, although [34,35] constructed two or more extended state observers to handle matched uncertainties and unmatched uncertainties respectively with adaption method, full-state information such as the velocity of the load and the load pressure of the cylinder in the proposed control strategies are needed to realize the state estimations, which, however, is a strict condition in many practical implementations, especially for hydraulic motion systems. In many cases, only position signals in hydraulic system are available and reliable because of structure constriction, cost limitation, measurement noisy, calculation errors, etc. [28,36–38], under these circumstances, full state feedback control is difficult to realize. It is these practical factors that make the advanced nonlinear control strategies unable to replace PID controller to dominate the hydraulic fields. Actually, whereas we know output feedback method is a powerful tool to handle these issues, very few progress about nonlinear output feedback is made for hydraulic system because of the complicated system information, unknown parametric uncertainties and modeling nonlinearities. Therefore, a natural thought comes to our mind: Can we use an output feedback method to realize adaptive control and ESO for a hydraulic system with a nonlinear friction model? If possible, it is reasonable to expect better tracking performance with less system information. To handle the aforementioned practical issues, this paper employs a continuous approximation of the LuGre friction model and proposes an extended state observer-based output feedback adaptive control for high performance motion control of hydraulic servo mechanism. The main advantages and contributions of this paper are summarized as follows: 1) An output feedback controller is constructed based on ESO with a parameter adaption law, which only relies on input signal and position output signal of the system. Besides, the synthesized adaptive

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Fig. 1. Schematic diagram of the double-rod hydraulic system.

law is driven by tracking errors and observation errors, thus the burden of ESO is alleviated and high gain feedback is avoided. 2) The employed friction model can preserve the main physical characteristics of the original model without increasing the complexity of the system stability analysis and can bring a continuous friction compensation to eliminate the majority of nonlinear parts. The proposed controller is more than just a simple combination of ESObased adaptive control and output feedback, infact, it is able to copes with both matched uncertainties and unmatched uncertainties with needing a little system information, which means its capability of rejecting uncertainties and disturbances from unknown parameters, signal pollution, and other unexpected dynamics is enhanced greatly. The Lyapunov-based analysis has proved this strategy presents a prescribed tracking transient performance and final tracking accuracy while obtaining asymptotic tracking performance in the presence of parametric uncertainties only. At last, comparative experiments are conducted on a hydraulic servo platform to verify the high tracking performance of the proposed control strategy. The rest parts of this paper are organized as follows. Problem formulation and dynamic models are described in Section 2. Section 3 gives the nonlinear output feedback controller design and its theoretical results. Section 4 presents comparative experimental results and analysis. Section 5 demonstrates the conclusions. 2. Problem formulation and dynamic models Fig. 1 illustrates the structure of the considered hydraulic system. As shown, a valve controlled double-rod hydraulic cylinder drives an inertia load for rectilinear motion. The control objective is to design an output-feedback control strategy to obtain satisfactory tracking performance for the hydraulic system. In Fig. 1, Q1 is the supplied flow rate of the forward chamber, and Q2 is the return flow rate of the return chamber; Ps is the supplied pressure of the fluid, and Pr is the return pressure. The dynamics of the inertia load can be written as my¨ = A p PL − fr + d

(1)

where m and y are the mass and the displacement of the load respectively; Ap represents the effective ram area of the cylinder; PL = P1 -P2 is load pressure between the two chambers

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of the actuator, where P1 and P2 are the pressures inside the two chambers of the actuator; fr is the nonlinear friction in the servomechanism and will be discussed later; ࢞d is viewed as the nonlinearities, including the unknown dynamics, external disturbance, as well as the approximation errors between the modeled friction fr and the real friction. The load pressure dynamics can be described by Vt P˙L = −A p y˙ − Ct PL + QL + m 4βe

(2)

where Vt is the total control volume of the actuator; β e is the effective oil bulk modulus; Ct is the coefficient of the total internal leakage of the actuator due to pressure; QL is the load flow. ࢞m represents uncertain nonlinearities caused by internal leakage, unknown pressure dynamics, modelling error of load flow, etc. QL is the function of spool valve displacement of the servo-valve xv , namely  QL = kq xv Ps − sgn (xv )PL (3) √ where kq = Cd W 1/ρ is the flow gain, Cd is the discharge coefficient; W is the spool valve area gradient; ρ is the density of oil, Ps is supplied pressure. sgn (∗ ) is a sign function defined by  if ∗ ≥ 0 1 sgn (∗ ) = (4) −1 if ∗ < 0 In some studies [39,40], researchers attempted to consider the dynamics of servo valve, however, the improvement of tracking performance is very limited. On the other hand, this consideration requires an additional sensor to measure the spool position and the design of controller becomes more complicated, therefore, this article does not consider the servo valve dynamics. In fact, the experimental analysis of this paper is based on a platform which utilizes a high-performance servo valve, hence the spool displacement can be described directly proportional to the control input voltage, i.e., xv = ki u, where ki is a positive constant, u is the control input voltage. So QL can be modeled by  QL = kt u Ps − sgn (u)PL (5) where kt = kq ki is the total flow gain with respect to u. There are three core variables (y, y˙, PL ) to be controlled for hydraulic motion systems, hence for simplification, the definition of the state variables is given by x = [x1 , x2 , x3 ]T = [y, y˙, A p PL /m]T . Thus the system model can be present in a state-space form as x˙1 = x2 x˙2 = x3 − f + d1 (t )

(6)

x˙3 = q(u, x3 )u + g(x2 , x3 ) + d2 4A β

where f = fr /m, d1 = md , d2 = mVp t e m  4A p βe kt m x3 q (u, x3 ) = Ps − sgn (u) mVt Ap   4A p βe m A p x2 + Ct x3 g(x2 , x3 ) = − mVt Ap

(7)

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3. Nonlinear output feedback controller design 3.1. Friction model As discussed before, LuGre model can capture the main frictional behaviors and meanwhile keep the properties of passivity and the boundness of internal states, however, its mathematical equations contain discontinuous terms so that it cannot be utilized to design the controller directly. By employing continuous and differentiable approximations of the discontinuous terms, Sobczyk et al. [20] proposed a new continuous extension of the LuGre model without affecting the passivity and boundness. Friction force fr is based on this continuous extension of the LuGre model and presented as follows fr = σa z + σb z˙ + σc y˙

(8)

where σ a is a stiffness coefficient, σ b is a damping coefficient related to z˙, σ c is a viscous friction coefficient. The model is the function of an average deflection z of the microscopic bristles of two contacting surfaces, the dynamic of z is given by z˙ = y˙S¯1 (y˙) −

S¯2 (y˙) z g(y˙)

(9)

where S¯1 (y˙) = [S0 (y˙)]2 and S¯2 (y˙) = y˙S0 (y˙) are two auxiliary functions, S0 (y˙) = 2 arctan (kv y˙)/π is one of the most common replacement function. g f (y˙) is a positive function defined as [16] g f (y˙) = Fc + (Fs − Fc )e−(y˙/y˙s )

2

(10)

where Fs and Fc are the static friction and Coulomb friction forces respectively, y˙s is the Stribeck velocity. Rewrite Eq. (8) as fr = σa z + σb z˙ + σc y˙S¯1 (y˙), then based on Eq. (9), the friction force fr can be arranged as fr = σa z + (σb + σc )y˙S¯1 (y˙) − σb

S¯2 (y˙) z g(y˙)

(11)

For simplicity, give the following definitions S1 (y˙) = y˙S¯1 (y˙), S2 (y˙) = S¯2 (y˙)/g(y˙) σa σb + σc σb σ0 = , σ1 = , σ2 = m m m Thus Eq. (9) can be transformed into z˙ = S1 (y˙) − S2 (y˙)z, and fr = σ0 z + σ1 S1 (y˙) − σ2 S2 (y˙)z m Then the estimation value of friction can be described by

(12)

f =

(13)

fˆ = σˆ 0 zˆ0 + σˆ 1 Sˆ1 (y˙) − σˆ 2 Sˆ2 (y˙)zˆ2

(14)

where σˆ i (i = 0, 1, 2) is the estimate of σi , Sˆi (i = 1, 2) is the estimate of Sˆi (y˙). Thus the estimation error of friction is f˜ = fˆ − f = σ˜ 0 zˆ0 + σ0 z˜0 + σ˜ 1 Sˆ1 − σ1 S˜1 − (σ˜ 2 Sˆ2 zˆ2 + σ2 Sˆ2 z˜2 − σ2 S˜2 z2 )

(15)

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where the estimation errors are defined by σ˜ i = σˆ i − σi , i = 0, 1, 2 z˜i = zˆi − zi , i = 0, 2 S˜i = Si − Sˆi , i = 1, 2

(16)

3.2. System model and issues to be addressed In this paper, the system model utilizes the nominal values of physical parameters (i.e., m, Ap , β e , kt , Vt , Ct , Ps ) in observers and controller design. The parameter errors are lumped to the unmodeled terms d1 (t) and d2 (t) in Eq. (6). Actually, since the friction model has captured the main part of nonlinear uncertainties, d2 (t) becomes the major modeling uncertainty to be solved. Extend d2 (t) as an additional state variable by defining x4 = d2 (t), then the system state x is extended to x = [x1 , x2 , x3 , x4 ]T . Define h(t) as the time derivative of x4 , thus the system state equations (6) can be rearranged as ⎧ x˙1 = x2 ⎪ ⎪ ⎪ ⎨x˙ = x − f + d (t ) 2 3 1 (17) ⎪ x ˙ = q(u, x ) u + g(x2 , x3 ) + x4 3 3 ⎪ ⎪ ⎩ x˙4 = h(t ) Before the design of control strategy, practical assumptions are made as follows: Assumption 1. The desired position trajectory yd = x1d (t) belongs to C3 and bounded. Assumption 2. The hydraulic system is working under normal condition, hence P1 and P2 are bounded by supplied pressure Pr and return pressure Ps , i.e., 0 ≤ Pr < P1 < Ps , 0 ≤ Pr < P2 < Ps , which ensures function q(u, x3 ) is far away from zero. Moreover, it is worth being noticed that q(u, x3 ) contains the sign function sgn(∗ ) so that it is not differentiable at u = 0. However, apart from this singular point (u = 0), q(u, x3 ) is always differentiable in other places and continuous everywhere, its left and right derivatives at u = 0 exist and finite [41]. Therefore, it makes sense for the following assumption to be proposed. Assumption 3. The function q(u, x3 ) is Lipschitz with respect to x3 and g(x2 , x3 ) is Lipschitz with respect to x2 and x3 in the practical range. 3.3. Projection mapping and parameter adaptation Practically, although we summarized the basic structure of the friction force, specific values of the parameters in friction model cannot be obtained exactly. In addition, in different working conditions, these parameters may be quite different since they are easily effected by temperature, relative velocity, wear condition and other unpredictable factors. To solve the parametric uncertainties, first of all, define ∗ i as the ith component of the vector ∗ , and the operation < for two vectors is performed in terms of the corresponding elements of the vectors. Besides, the definitions of σ , σˆ and σ˜ can be found in Eq. (16). Hence, a discontinuous projection can be defined as [25,33]. ⎧ ⎨ 0 if σˆ i = σi max and ∗ i > 0 Proj(∗ i ) = 0 if σˆ i = σi min and ∗ i < 0 (18) ⎩∗ otherwise i

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By employing an adaptation law given by σ˙ˆ = Projσˆ (τ ) with σmin ≤ σˆ (0 ) ≤ σmax

(19)

where Г is a positive diagonal adaption matrix while τ is an adaption function to be synthesized later; hence with the projection mapping used in Eq. (19), the proposed adaptive algorithm guarantees  (20) σˆ ∈ σˆ = σˆ : σmin ≤ σˆ ≤ σmax On the other hand, accurate value of the friction state z in Eq. (13) cannot be measured as well. Therefore, we construct a state observer to estimate the unmeasurable state z. To cope with the different characteristics of z, based on a dual-observer structure is utilized with projection mappings [13,19] z˙ˆ0 = Projzˆ (χ0 ), z˙ˆ2 = Projzˆ (χ2 ) (21) 0

2

where zˆ0 and zˆ2 are estimates of the unmeasurable friction state z; χ 0 and χ 2 are the adaption functions which will be designed later. The same as the projection mapping proposed in Eq. (18), we have ⎧ ⎪ ⎨ 0 if zˆi = zmax and χi > 0 Projzˆ i (χi ) = 0 if zˆi = zmin and χi < 0 (22) ⎪ ⎩χ otherwise i where zmax and zmin are the upper bound and lower bound of the friction internal state z respectively, which are used to restrict the observation values of state z. From Eqs. (9) and (10), it can be inferred that the observation bounds can be set as zmax = Fs , zmin = −Fs [13]. Similarly, the proposed projection mapping guarantees zi min ≤ zˆi ≤ zi max , i = 0, 2

(23)

3.4. Extend state observer design In this section, observers are designed to observe the unmeasured system states (x2 , x3 ) but as well as the modeling uncertainty x4 for controller design. Define x = [x1 , x2 , x3 , x4 ]T , xˆ as the estimate of x and x˜ = x − xˆ as the estimation error; y = Cx = x1 , where C = [1,0,0,0]. Based on the extended system model (17), the linear extended state observers (LESO) [33] are then constructed as ⎧

 ˙1 = xˆ2 + 4w0 x1 − xˆ1 ⎪ x ˆ ⎪ ⎪ ⎪ ⎨x˙ˆ = xˆ − fˆ + 6w2 x − xˆ  2 3 1 0 1 (24)





 ˙3 = q u, xˆ3 u + g xˆ2 , xˆ3 + xˆ4 + 4w3 x1 − xˆ1 ⎪ x ˆ ⎪ 0 ⎪

 ⎪ ⎩˙ xˆ4 = w04 x1 − xˆ1 where ω0 >0 is the only tuning parameter of the observer, which can be viewed as the bandwidth of the observer. According to Eqs. (17) and (24), the dynamics of the state estimation errors can be written as ⎧˙ x˜1 = x˜2 − 4w0 x˜1 ⎪ ⎪ ⎪ ⎨x˜˙ = x˜ + f˜ + d − 6w2 x˜ 2 3 1 0 1 (25) 3 ˙˜3 = x˜4 + qu ⎪ x ˜ + g ˜ − 4w ⎪ 0 x˜1 ⎪ ⎩˙ x˜4 = h − w4 x˜1 0

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where q˜ = q(u, x3 ) − q(u, xˆ3 ), g˜ = g(x2 , x3 ) − g(xˆ2 , xˆ3 ). Define ε = [ε1 , ε2 , ε3 , ε4 ]T = [x˜1 , x˜2 /w0 , x˜3 /w02 , x˜4 /w03 ]T , then Eq. (25) can be rearranged as ε˙ = w0 Aε + B + D + B2 φ/w0

(26)

where φ = σ˜ 0 zˆ0 + σ0 z˜0 + σ˜ 1 Sˆ1 − σ˜ 2 Sˆ2 zˆ2 − σ2 Sˆ2 z˜2 , ⎡ ⎤ ⎡ ⎤ 0 −4 1 0 0 ⎢−6 0 1 0⎥ ⎢(−σ1 S˜1 + σ2 S˜2 z2 )/w0 ⎥   ⎥ ⎢ ⎥, D = 0, d1 /w0 , 0, h/w3 T , A=⎢ 0 ⎣−4 0 0 1⎦, B = ⎣ ⎦ (qu ˜ + g˜)/w02 −1 0 0 0 0 B2 = [0, 1, 0, 0]T

(27)

Note that matrix A is Hurwitz, so there exists a positive definite matrix Pε satisfying the following equation AT Pε + Pε A = −I

(28) 1 T ε Pε ε, 2

Considering a basic Lyapunov function Vε = its time derivative expression can be written as     1 φ T 1 T φ T T T T ˙ w0 ε A + B + D + Vε = B Pε ε + ε Pε w0 Aε + B + D + B2 2 w0 2 2 w0

 1 φ T = w0 ε T AT Pε + Pε A ε + BT Pε ε + DT Pε ε + B Pε ε (29) 2 w0 2 By borrowing the theorem assumption B2T Pε = κC [42], where κ = 0, then Eq. (29) can be transformed into φ 1 V˙ε = − w0 ε 2 + BT Pε ε + DT Pε ε + κCε (30) 2 w0 Based on the expressions of Eqs. (26) and (30), from the stabilizing analysis method about high-gain observers [43], we can conclude that the designed LESO in Eq. (24) is stable and the state estimation error can be regulated arbitrarily small by increasing the bandwidth ω0 . The closed loop stability of the system will be proved later in great detail. 3.5. Nonlinear backstepping controller design Backstepping design [36,44] is widely employed in the controller design to cope with the unmatched uncertainties. At first, a switching-function-like quantity is defined as e e2 = e˙1 + k1 e1 = x2 − x2eq , x2eq = x˙1d − k1 e1

(31)

where e1 = x1 -x1 d (t) is the output tracking error; k1 is a positive feedback gain; x2 eq is a measurable variable while e2 is not measurable because it relies on the unmeasured state x2 . Based on Eq. (17), the time derivative of e2 can be written as e˙2 = x˙2 − x˙2eq = x3 − f + d1 − x¨1d + k1 x2 − k1 x˙1d

(32)

In Eq. (29), x3 is regarded as a virtual control input which will be synthesized later. To help design x3 , define a virtual control law α 2 for x3 and then make e3 = x3 -α 2 denote the

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input discrepancy. Then by using Eq. (32) and the state estimation (24), the resulting virtual control law α 2 can be designed as α2 = α2a + α2s α2a = fˆ + x¨1d − k1 xˆ2 + k1 x˙1d

 α2s = −k2 xˆ2 − x2eq = −k2 e2 + k2 x˜2

(33)

where k2 is a positive feedback gain, α 2 a is treated as a model-based compensation term via online state estimation while α 2 s is designed as a robust control law to stabilize the uncertainties of the nonlinear system. Combining Eqs. (32) and (33), one obtains e˙2 = e3 − k2 e2 + k1 x˜2 + k2 x˜2 + f˜ + d1

(34)

After designing the virtual control law α 2 , an actual control law for u will be synthesized in this part. According to Eq. (17), we can obtain the time derivative of e3 , namely e˙3 = x˙3 − α˙ 2 = q(u, x3 )u + g(x2 , x3 ) + x4 − α˙ 2c − α˙ 2u

(35)

where ∂ α2 ∂ α2 ˙ ∂ α2 ˙ ∂ α2 ˙ ∂ α2 ∂ α2 ˙ + σˆ + zˆ1 + zˆ2 + xˆ2 + xˆ2 ∂t ∂ σˆ ∂ zˆ1 ∂ zˆ2 ∂ x1 ∂ xˆ2 ∂ α2 = x˜2 ∂ x1

α˙ 2c = α˙ 2u

(36)

Actually, the time derivative of α 2 is divided into two parts, where α˙ 2c denotes the calculable part which can be utilized when designing the controller while α˙ 2u is the incalculable one because of the unmeasurable state. Based on Eqs. (35) and (36), the actual control input is designed as  1 (37) u = −g(xˆ2 , xˆ3 ) − xˆ4 + α˙ 2c − k3 (xˆ3 − α2 ) qˆ where qˆ is the estimation of q(u, x3 ) and is always positive. Note that xˆ3 − α2 = x3 − x˜3 − α2 = e3 − x˜3 , by substituting Eq. (37) into Eq. (35), one obtains ∂ α2 e˙3 = −k3 e3 + k3 x˜3 + qu ˜ + g˜ + x˜4 − x˜2 (38) ∂ x1 For clearer presentation, a block diagram of the developed control strategy is summarized in Fig. 2. 3.6. Main results To facilitate the subsequent analysis, a set of scalars are defined below  = 21 w0 − λmax (Pε )CB , CS = σ1 w0CS1 + σ2 w0CS2 zmax ,     γ1 = w0 (k1 + k2 ) + CS , γ2 = Cg1 +  ∂∂αx12 , γ3 = Cg2 + k3 w02 + Cq |umax |, γ4 = w03 , δ ≥ σM  · ϕ0  + σ0 zM + σ2 zM Sˆ2 + |d1 |max ,   ζ = λmax (Pε )D · ε  + σM  · ϕ1  + κwε01  zM σ0 + σ2 Sˆ2 + ηδ 2

(39)

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Fig. 2. Block diagram of the developed control strategy.

Note Assumption 2 and from the definitions of S1 (x2 ), S2 (x2 ), g(x2 , x3 ) as well as ε, there exists a set of known constants CS 1 , CS 2 , Cg 1 and Cg 2 so that S˜1 = S1 (x2 ) − S1 (xˆ2 ) ≤ CS1 x˜2 S˜2 = S2 (x2 ) − S2 (xˆ2 ) ≤ CS2 x˜2 g˜ = g(x2 , x3 ) − g(xˆ2 , xˆ3 ) ≤ Cg1 ε2 + Cg2 ε3

(40)

In addition, notice the expression of B(u, x2 , x3 ) in (27), we can know B ≤ CB ε 

(41)

where CB is a known constant. Theorem 1. In the presence of parametric uncertainties only, i.e., x2 = xˆ2 , d1 = 0, d˙2 = 0, with the projection type adaptation law (19) and then the adaptation function τ is determined as τ = ϕ1 + ϕ2 where T T

 κCε  ϕ1 = zˆ0 , Sˆ1 , −Sˆ2 zˆ2 , ϕ2 = xˆ2 − x2eq zˆ0 , Sˆ1 , −Sˆ2 zˆ2 w0

(42)

(43)

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zˆ0 and zˆ2 can be obtained from Eq. (21), the adaption functions χ 0 and χ 2 are synthesized to be

 χ0 = Sˆ1 − Sˆ2 zˆ0 − β0 κCε/w0 − β0 xˆ2 − x2eq

 (44) χ2 = Sˆ1 − Sˆ2 zˆ2 + β2 Sˆ2 κCε/w0 + β2 Sˆ2 xˆ2 − x2eq By choosing proper positive coordinating parameters ω0 and large enough feedback gains k1 , k2 , and k3 , the matrix  defined below can be treated as positive definite ⎡ ⎤ ⎡ ⎤ a 0 c k1 −1/2 0  0 ⎦, a = ⎣−1/2 k2 −1/2⎦, =⎣ 0 Tc 0 b 0 −1/2 k3 ⎡ ⎤ ⎡ ⎤  0 0 0 0 0 0 ⎦, c = ⎣−γ1 /2 0 0 ⎦ b = ⎣ 0  (45) 0 0  −γ2 /2 −γ3 /2 −γ4 /2 where 0 denotes zero vector with compatible dimensions. Hence functioned by the designed control law (37), all closed loop system signals are guaranteed to be bounded and the output tracking is guaranteed to be asymptotic, namely, e1 →0 as t→∞. Proof. See Appendix A. Remark 1. Results of Theorem 1 demonstrate that if hydraulic system is under the circumstance with only parametric uncertainties and constant uncertain nonlinearities, thus the constant uncertain nonlinearities can be exactly estimated by the constructed ESO and then compensated by the synthesized controller via feed-forward method. Moreover, the proposed nonlinear output feedback controller has the capability of adjustable converging rate by adjusting the feedback gains k1 , k2 and k3 properly, thus, desirable tracking error can be achieved. Additionally, asymptotic tracking performance can be obtained as well, which is of importance in the high performance tracking control of hydraulic implementations. Theorem 2. In other working condition that the system contains modelling errors, i.e., d1 = 0, d˙2 = 0, the tracking error e defined by e = [e1 , e2 , e3 ] and the scaled state estimation error ε are bounded. Define a positive definite function V2 as follows 1 V2 = e2 + Vε (46) 2 V2 is bounded by  ζ  1 − exp(−λ2t ) (47) V2 (t ) ≤ V2 (0) exp(−λ2t ) + λ2 where λ2 = 2λmin (2 )min{1,1/λmax (Pε )}, λmin (·) and λmax (·) are the minimum and maximum eigenvalues of the matrix respectively. Proof. See Appendix B. Remark 2. Results of Theorem 2 indicate that in the situation with both parametric uncertainties and time-variant uncertain nonlinearities, the tracking error and state estimation errors are guaranteed to be bounded. Moreover, according to Eq. (47) we know that the transient performance and final tracking error can be improved by adjusting certain controller parameters. In other words, by increasing the robust feedback gains, the transient performance can be

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C. Luo, J. Yao and J. Gu / Journal of the Franklin Institute 356 (2019) 8414–8437 Table 1 Specifications of the EHLS and aircraft actuation system. Components

Specifications

Hydraulic supply Servo valve

Supply pressure Type Rated flow Bandwidth Stroke Efficient ram area Mass Type Accuracy Type Accuracy Type Type Type Type

Hydraulic actuator Load mass Linear encoder Pressure sensors A/D card D/A card Counter card Computer

120 bar Moog G761-3003 19 L/min at 70 bar drop ≥120 Hz 44 mm 904.78 mm2 30 kg Heidenhain LC483 ±5 μm MEAS US175-C00002-200BG 1 bar Advantech PCI-1716 Advantech PCI-1723 Heidenhain IK-220 IEI WS-855GS

Fig. 3. Experimental equipment of double-rod hydraulic actuator system.

enhanced and the final tracking error ζ /λ2 can be made as small as possible. Most importantly, in addition to increasing the robust feedback gains, the final tracking error can be limited by adjusting the observer gain and then ζ can be decreased. By doing so, many problems from high gain feedback would be avoided. 4. Comparative experimental results 4.1. Experiment setup In order to verify the proposed design, an experimental platform with a double-rod hydraulic actuator is set up and shown in Fig. 3. This platform includes a bench case, a hydraulic cylinder, a linear encoder and a high bandwidth servo valve. In addition, there are a set of inertial steel sheets, a hydraulic supplier, a set of measurement and control software. Specifications of these hardware components are listed in Table 1. The measurement and control software are made up of a monitor software and a real time control software. The monitor software is programmed with NI

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LabWindows/CVI, and the real time control software is compiled with Microsoft visual studio 2005 plus Ardence RTX 7.0, which is utilized to offer a real-time environment for real time control software under Windows XP operating system. Additionally, the sampling time T is 0.5 ms. 4.2. Comparative results To verify the effectiveness and feasibility of the designed control algorithm, three controllers are compared as follows. (1) OFAEF: This is the proposed output feedback adaptive controller with ESO and LuGre friction compensation developed in this paper and described in Section 3. Feedback gains k1 , k2 , k3 were chosen large enough so that the system stability can be guaranteed, hence the following feedback gains: k1 = 800, k2 = 450, k3 = 185. The observer gains are: ω0 = 200. To prove the convergence of the parameters under parameter adaptation law, all estimates of θ begin at zero. The bounds of θ are given as: σ min = [0, 0, 0]T ; σ max = [2 × 104 , 40, 1000,]T . The parameter adaptation rates are set at Г = diag{3 × 1013 , 6000, 8 × 1015 }, β 0 = 1 × 10−4 , β 2 = 1 × 10−4 . (2) ESOC: This is the ESO based output feedback controller which was proposed by Han [30]. The state variables are defined as x = [x1 , x2 , x3 ]T = [y, y˙, A p PL /m]T . The main information of this controller is shown as follows:  



 4βe A2 xˆ2 + Ct mxˆ3 1  u= − xˆ4 + α˙ 2c − v3 xˆ3 − α2 , mVt q u, xˆ3

 4000xˆ2 + 100a tan (900x˙1d ) + x¨1d − v1 xˆ2 + v1 x˙1d − v2 xˆ2 − x2eq , m ∂ α2 ∂ α2 ∂ α2 ˙ α˙ 2c = + xˆ2 + xˆ2 , x2eq = x˙1d − v1 z1 , z1 = x1 − x1d , ∂t ∂ x1 ∂ xˆ2



 x˙ˆ = A0 xˆ +  xˆ + Gu + x1 − xˆ1

(48)

in which   

 4000xˆ2 + 100a tan (900x˙1d )  xˆ = 0, − , g xˆ2 , xˆ3 , 0 , m 

 T  T G = 0, 0, q u, xˆ3 , 0 , H = 4w0 , 6w02 , 4w03 , w04

(49)

α2 =

The control parameters of this controller are given by v1 = 800, v2 = 450, v3 = 185, w0 = 200. (3) VFPI: This is the velocity feed-forward proportional-integral (PI) controller, which is widely applied in industrial implementations. The control input u of this controller can be described by u = kP e1 (t ) + kI

e1 (t )dt + x˙d /Kv

(50)

where e1 (t) = x1 –x1d is the tracking error, kP and kI are proportional gain and integral gain respectively, Kv is chosen as the open loop velocity gain, and kP = 8000, kI = 2000, Kv = 35.55. Remark 3. The main difference between OFAEF and ESOC is OFAEF replaces the traditional friction model in ESOC controller with a continuous LuGre friction model, which is utilized

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C. Luo, J. Yao and J. Gu / Journal of the Franklin Institute 356 (2019) 8414–8437 Table 2 Performance indexes for 0.5 Hz. Indexes

Me

μ

σ

OFAEF ESOC VFPI

0.0458 0.0572 0.0529

0.0141 0.0291 0.0271

0.0103 0.0142 0.0152

to verify that the nonlinear friction effects in hydraulic servo systems can be effectively suppressed by the continuous LuGre model-based friction compensation scheme. Besides, the convergence of parameter estimation of OFAEF can be also verified since all parameters in ESOC are known. Furthermore, to quantify the experimental results of each controller, three performance indexes are introduced in this paper, i.e., the maximum, average, and standard deviation of the tracking errors. Their definitions are explained as follows. (1) Maximal absolute value of the tracking errors Me = max {|z1 (i)|} i=1,...,N

(51)

where N is the number of the recorded digital signals, and is used as an index of measure of tracking accuracy. (2) Average value of the tracking errors μ=

N 1 ! |z1 (i)| N i=1

(52)

and is used as an objective numerical measure of average tracking performance. (3) Standard deviation value of the tracking errors " # N #1 ! $ σ = [|z1 (i)| − μ]2 N i=1

(53)

to measure the deviation level of tracking errors. To investigate the tracking performance of the proposed control algorithms, the experimental command is set as a sinusoidal-like trajectory. The desired position trajectory is xd = 10arctan(sin(π t))[1-e−t ]/0.7854 mm. As is shown in Fig. 4, the tracking results of three controllers are illustrated. Besides, Table 2 demonstrates their performance indexes during the last two cycles to explore the final tracking performance. Visually, from Fig. 4 we can see that the tracking curve of OFAEF controller is apparently superior to the ones of other two controllers. Moreover, this observation is also proved by the mathematical calculation shown in Table 2. From the comparative results between OFAEF and ESOC, it can be seen that the proposed OFAEF controller achieved better tracking performance among all performance indexes, which means the modified LuGre model-based friction compensation scheme is very helpful to alleviate the effects from nonlinear friction in hydraulic motion systems. On the

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Fig. 4. Tracking performance of the three controllers for 0.5 Hz.

Fig. 5. Parameter estimation of OFAEF controller for 0.5 Hz.

other hand, obviously the tracking performance of VFPI is inferior to OFAEF. In fact, VFPI made similar achievement with ESOC among these three indexes, this is because in such working cases, the hydraulic system mainly appears integral characteristic, by using velocity feedforward compensation, the unknown dynamics are compensated and the other uncertainties can be easily suppressed via strong feedback gains. Fig. 5 shows the parameter estimation of OFAEF controller. To explore the capability of parameter estimation, we manually and deliberately chose the initial estimation values of parameters far away from their optimal values. As seen in Fig. 5, because of the large initial parameter errors, the transient performance of OFAEF controller is not very well, but after a short time, excellent final tracking performance are obtained with the assistance of parameter estimation. Additionally, Fig. 5 demonstrates

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Fig. 6. Control input of OFAEF controller for 0.5 Hz.

Fig. 7. Tracking errors of three controllers for 0.2 Hz.

that the parameter estimations of OFAEF controller can converge to their actual values after a period of adjustment. As a result, the effectiveness of the parameter adaption is verified. The control input of the proposed OFAEF controller is presented in Fig. 6, it can be seen that the control input is bounded and continuous, which is very significant and practical in industrial implementations. It is worth noting that if stricter transient tracking performance is required in practical implementations, the initial estimation values of parameters are suggested to be chosen on the basis of their empirical values other than 0. For more investigations about the tracking ability of the proposed controller in different working conditions, experiments are run for another sinusoidal-like command. The desired motion trajectory is set by xd = 10arctan(sin(0.4π t))[1-e−t ]/0.7854 mm. Fig. 7 shows the tracking performance of three controllers and Table 3 collects their performance indexes

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Table 3 Performance indexes for 0.2 Hz. Indexes

Me

μ

σ

OFAEF ESOC VFPI

0.0282 0.0335 0.0286

0.0083 0.0133 0.0134

0.0055 0.0093 0.0080

Fig. 8. Tracking errors of three controllers for 0.1 Hz.

during the last two cycles. In this experimental condition, both unmodeled disturbances and nonlinear friction force influence the tracking capability greatly. According to these test results, it is obvious that the synthesized OFAEF controller is superior to the other two controllers even under such a complicated tracking circumstance. This is because ESOC controller cannot solve the complicated friction dynamic while VFPI controller cannot alleviate the unknown uncertainties effectively. Although the difference between OFAEF and VFPI on index Me is not big, much smaller average tracking error μ and standard deviation σ imply that OFAEF controller can achieve better steadiness and anti-vibration performance. Practically, for a slow motion condition, nonlinear friction becomes the dominating factor influencing the tracking performance, because in this case, the nonlinear friction force is mainly concentrated on the Stribeck effect area. Therefore, in the end, a slow desired motion trajectory command xd = 10arctan(sin(0.2π t)) [1-e − t ]/0.7854 mm is set to investigate the tracking performance in this typical test. Under the circumstances, the experimental results of three controllers are illustrated in Fig. 8. As shown, compared with other two controller, the proposed OFAEF controller still obtains the best tracking performance as before even for such a challenging tracking case. As a result, it is reasonable to believe the proposed OFAEF controller can cope with the nonlinear friction and attenuate the unmodeled dynamics to an acceptable range. In addition, corresponding performance indexes during the last two cycles are collected in Table 4. It also

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C. Luo, J. Yao and J. Gu / Journal of the Franklin Institute 356 (2019) 8414–8437 Table 4 Performance indexes for 0.1 Hz. Indexes

Me

μ

σ

OFAEF ESOC VFPI

0.0040 0.0062 0.0048

0.000753 0.0013 0.0010

0.000629 0.0010 0.00878

can be concluded from these results that OFAEF controller can achieve excellent tracking performance among all the indexes. 5. Conclusion In this paper, an extended state observer-based output feedback adaptive controller with a continuous LuGre friction compensation is proposed for a hydraulic servo control system. To handle the nonlinear friction dynamics in various complicated working conditions, a continuous approximation of the LuGre friction model is investigated. This friction model can preserve the major physical characteristics of the original model without increasing the complexity of the system stability analysis and thus a continuous friction compensation is obtained to eliminate the majority of nonlinear parts. More importantly, an extended state observer with new parameter adaption law is constructed, and then both matched uncertainties and unmatched uncertainties are solved by feed-forward compensation. In addition, output feedback is realized on the basis of the proposed extended state observer adaptive controller, which can reduce the effects caused by signal pollution, measurement noise, ect. The synthesized controller not only guarantees a prescribed tracking transient performance and final tracking accuracy but also obtains asymptotic tracking performance in the presence of parametric uncertainties only, which is proved via Lyapunov-based analysis. In the end, comparative experiments are conducted on a hydraulic servo platform to verify the high tracking performance of the proposed control strategy. Though this paper considers nonlinear friction dynamic of the hydraulic system and apparently improves the tracking performance, there are lots of other nonlinear factors in hydraulic systems. Future work should center on problems of dead-zone and time-delay. Furthermore, the main thought of the proposed control strategy could be combined with other control objectives such as hydraulic robotics, machine tools and so on. We believe this practical control strategy would contribute to achieving better control performance in relevant implementations. Acknowledgment C. Luo is grateful to Prof. J. Gu for his guidance and host at Dalhousie University from September 2017. This work was supported in part by the National Natural Science Foundation of China under Grant 51675279, in part by the National Key Research and Development Program of China under Grant 2018YFB2000702, in part by the Natural Science Foundation of Jiangsu Province under Grant BK20170035, in part by Joint Fund of the Ministry of Education of China (6141A020331) and in part by the Fundamental Research Funds for the Central Universities under Grant 309171A8801.

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Appendix A

Proof. of Lemma 1. In this case, define the following Lyapunov function V =

1 2 1 T 1 1 1 e + ε Pε ε + σ˜ T  −1 σ˜ + σ0 β0−1 z˜02 + σ2 β2−1 z˜22 2 2 2 2 2

(A1)

According to the dynamics of e, ε, σ , z0 , and z2 described in Eqs. (19), (21), (26), (31) and (35), one obtains V˙ = − k1 e21 + e1 e2 − k2 e22 + e2 e3 + (k1 + k2 )w0 e2 ε2 + e2 f˜ + e2 d1 − k3 e23 + k3 w02 e3 ε3 ∂ α2 1 + e3 qu ˜ + e3 g˜ + w03 e3 ε4 − w0 e3 ε2 − w0 ε 2 + BT Pε ε + DT Pε ε + σ˜ T φ1 ∂ x1 2  κCε  σ0 z˜0 − σ2 Sˆ2 z˜2 + σ˜ T  −1 σ˙ˆ + σ0 β0−1 z˜0 z˙˜0 + σ2 β2−1 z˜2 z˙˜2 + w0 where ϕ1 =

(A2)

κCε [zˆ0 , Sˆ1 , −Sˆ2 zˆ2 ]T . w0

Note that e2 = xˆ2 − x2eq + x˜2 and e2 f˜ = e2 (σ˜ 0 zˆ0 + σ0 z˜0 + σ˜ 1 Sˆ1 − σ1 S˜1 − σ˜ 2 Sˆ2 zˆ2 − ˆ σ2 S2 z˜2 + σ2 S˜2 z2 ) then Eq. (A2) can be written by

1 V˙ = − k1 e21 − k2 e22 − k3 e23 − w0 ε 2 + e1 e2 + e2 e3 + (k1 + k2 )w0 e2 ε2 2   ∂ α2 + e2 −σ1 S˜1 + σ2 S˜2 z2 + k3 w02 e3 ε3 + e3 qu ˜ + e3 g˜ + w03 e3 ε4 − w0 e3 ε2 + BT Pε ε ∂ x1  κCε  σ0 z˜0 − σ2 Sˆ2 z˜2 + σ˜ T  −1 σ˙ˆ + σ0 β0−1 z˜0 z˙˜0 + σ2 β2−1 z˜2 z˙˜2 + DT Pε ε + σ˜ T ϕ1 + w0   

 + xˆ2 − x2eq σ0 z˜0 − σ2 z˜2 Sˆ2 + e2 d1 + σ˜ T ϕ2 + x˜2 σ˜ 0 zˆ0 + σ˜ 1 Sˆ1 − σ˜ 2 Sˆ2 zˆ2 + σ0 z˜0 − σ2 Sˆ2 z˜2 (A3) where ϕ2 = (xˆ2 − x2eq )[zˆ0 , Sˆ1 , −Sˆ2 zˆ2 ]T . Then based on the inequalities (40) and (41), we have 1 V˙ ≤ − k1 e21 − k2 e22 − k3 e23 − w0 ε 2 + e1 e2 + e2 e3 + (k1 + k2 )w0 e2 ε2 + σ1 w0CS1 e2 ε2 2 + σ2 w0CS2 zmax e2 ε2 + k3 w02 e3 ε3 + Cg1 e3 ε2 + Cg2 e3 ε3 + Cq |umax |e3 ε3 + w03 e3 ε4      ∂ α2   w0 e3 ε2 + λmax (Pε )CB ε 2 + σ˜ T  −1 σ˙ˆ + ϕ1 + ϕ2 + ∂ x1  

 + σ0 z˜0 β0−1 z˙ˆ0 − z˙0 + β0 κCε/w0 + β0 xˆ2 − x2eq 

 + σ2 z˜2 β2−1 z˙ˆ2 − z˙2 − β2 Sˆ2 κCε/w0 − β2 Sˆ2 xˆ2 − x2eq   + e2 d1 + ε T Pε D + x˜2 σ˜ 0 zˆ0 + σ˜ 1 Sˆ1 − σ˜ 2 Sˆ2 zˆ2 + σ0 z˜0 − σ2 Sˆ2 z˜2

(A4)

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In Eq. (A4), 

 σ0 z˜0 β0−1 z˙ˆ0 − z˙0 + β0 κCε/w0 + β0 xˆ2 − x2eq   

 = σ0 z˜0 β0−1 z˙ˆ0 − Sˆ1 + Sˆ2 zˆ0 + β0 κCε/w0 + β0 xˆ2 − x2eq + σ0 z˜0 β0−1 −S˜1 + S˜2 zˆ0 − σ0 z˜02 β0−1 S2

(A5)

Similarly, 

 σ2 z˜2 β2−1 z˙ˆ2 − z˙2 − β2 Sˆ2 κCε/w0 − β2 Sˆ2 xˆ2 − x2eq   

 = σ2 z˜2 β2−1 z˙ˆ2 − Sˆ1 + Sˆ2 zˆ2 − β2 Sˆ2 κCε/w0 − β2 Sˆ2 xˆ2 − x2eq + σ2 z˜2 β2−1 −S˜1 + S˜2 zˆ2 − σ2 z˜22 β2−1 S2

(A6)

Hence the dual-observer structure estimates of the unmeasurable friction state z (21) and (44). Define E = [e1 , e2 , e3 , ε1 , ε2 , ε3 , ε4 ]T = [eT , εT ]T and note that only parametric uncertainties exist in this case, namely x2 = xˆ2 , d1 = 0, d˙2 = 0, Hence, V˙ ≤ − E T E − σ0 z˜02 β0−1 S2 − σ2 z˜22 β2−1 S2

 ≤ − E T E ≤ −λmin () e2 + ε 2 = W

(A7)

Therefore, W∈L2 and V∈L∞ . Since all states are bounded and according to Eqs. (26), (34) and (35), it can be inferred that W˙ is also bounded and uniformly continuous. Via Barbalat’s lemma, W→0 as t→∞ [45], hence the result of Theorem 1. Appendix B Proof. of Theorem 2. Provided the uncertain nonlinearities are time-variant, i.e., d1 = 0, d˙2 = 0, then the time derivative of Eq. (46) is given by V˙2 = − k1 e21 + e1 e2 − k2 e22 + e2 e3 + (k1 + k2 )w0 e2 ε2 + e2 f˜ + e2 d1 − k3 e23 + k3 w02 e3 ε3 + qu ˜ + e3 g˜   κCε ∂ α2 1 σ0 z˜0 − σ2 Sˆ2 z˜2 + w03 e3 ε4 − w0 e3 ε2 − w0 ε 2 + BT Pε ε + DT Pε ε + σ˜ T ϕ1 + ∂ x1 2 w0 (B1) Define k2 = k2a + k2b and rearrange Eq. (B1), one obtains 1 V˙2 ≤ − k1 e21 − k2a e22 − k3 e23 − w0 ε 2 + e1 e2 + e2 e3 + (k1 + k2 )w0 e2 ε2 + k3 w02 e3 ε3 2    ∂ α2  3  w0 e3 ε2 + λmax (Pε )CB ε 2 + Cg1 e3 ε2 + Cg2 e3 ε3 + Cq |umax |e3 ε3 + w0 e3 ε4 +  ∂ x1     κε1   σ0 zM + σ2 Sˆ2 zM + e2 f˜ + d1 − k2b e2 + λmax (Pε )D · ε  + σM  · ϕ1  + w0 (B2) where σM = σmax − σmin , zM = zmax − zmin .

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Notice   e2 f˜ + d1 − k2b e2   = e2 σ˜ 0 zˆ0 + σ0 z˜0 + σ˜ 1 Sˆ1 − σ1 S˜1 − σ˜ 2 Sˆ2 zˆ2 − σ2 Sˆ2 z˜2 + σ2 S˜2 z2 + d1 − k2b e2   ≤ e2 σ˜ T ϕ0 + σ0 z˜0 − σ2 Sˆ2 z˜2 + d1 − k2b e2 + σ1 w0CS1 e2 ε2 + σ2 w0CS2 zmax e2 ε2

(B3)

If δ ≥ σM  · ϕ0  + σ0 zM + σ2 zM Sˆ2 + |d1 |max and choose k2 b = 1/(4η), then we have   e2 e2 f˜ + d1 − k2b e2 ≤ − 2 + e2 δ + CS e2 ε2 4η  2 e2 √ =− √ − ηδ + ηδ 2 + CS e2 ε2 2 η ≤ηδ 2 + CS e2 ε2

(B4)

where CS = σ1 w0CS1 + σ2 w0CS2 zmax . Hence Eq. (B4) can be written as 1 V˙2 ≤ − k1 e21 − k2a e22 − k3 e23 − w0 ε 2 + λmax (Pε )CB ε 2 + e1 e2 + e2 e3 + (k1 + k2 )w0 e2 ε2 2    ∂ α2   w0 e3 ε2 + k3 w2 e3 ε3 + Cg2 e3 ε3 + Cq |umax |e3 ε3 + w3 e3 ε4 + CS e2 ε2 + Cg1 e3 ε2 +  0 0 ∂ x1   κε1   σ0 zM + σ2 Sˆ2 zM + ηδ 2 + λmax (Pε )D · ε  + σM  · ϕ1  + w0 T (B5) ≤ − E 2 E + ζ where

⎡

⎤ −1/2 0 k2a −1/2⎦ −1/2 k3   κε1  ζ = λmax (Pε )D · ε  + σM  · ϕ1  + zM σ0 + σ2 Sˆ2 + ηδ 2 w0

a2 2 = ⎣ 0 Tc

0  0

⎤ ⎡ c k1 0 ⎦, a2 = ⎣−1/2 b 0

(B6)

According to Comparison Lemma [42], then Eq. (47) is obtained. Consequently, e and ε are bounded, that is to say, state x and its estimation are bounded as well. In the end, results of Theorem 2 is proved. References [1] Y. Yang, L. Ma, D. Huang, Development and repetitive learning control of lower limb exoskeleton driven by electrohydraulic actuators, IEEE Trans. Ind. Electron. 64 (6) (2017) 4169–4178. [2] W. Sun, H. Gao, B. Yao, Adaptive robust vibration control of full-car active suspensions with electrohydraulic actuators, IEEE Trans. Control Syst. Technol. 21 (6) (2013) 2417–2422. [3] S. Wu, Z. Jiao, L. Yan, W. Dong, A high flow rate and fast response electrohydraulic servo valve based on a new spiral groove hydraulic pilot stage, ASME J. Dyn. Syst. Meas. Control 137 (6) (2015) 061010. [4] J. Koivumäki, J. Mattila, Stability-guaranteed force-sensorless contact force/motion control of heavy-duty hydraulic manipulators, IEEE Trans. Robot. 31 (4) (2015) 918–935. [5] C. Luo, J. Yao, F. Chen, L. Li, Q. Xu, Adaptive repetitive control of hydraulic load simulator with rise feedback, IEEE Access 5 (2017) 23901–23911.

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