Finite-time adaptive sliding mode force control for electro-hydraulic load simulator based on improved GMS friction model

Mechanical Systems and Signal Processing 102 (2018) 117–138

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Finite-time adaptive sliding mode force control for electrohydraulic load simulator based on improved GMS friction model Shuo Kang a, Hao Yan a,b,⇑, Lijing Dong a,b, Changchun Li a,b a b

College of Mechanical Electronic and Control Engineering, Beijing JiaoTong University, Beijing 100044, China Key Laboratory of Vehicle Advanced Manufacturing, Measuring and Control Technology (Beijing JiaoTong University), Ministry of Education, Beijing 100044, China

a r t i c l e

i n f o

Article history: Received 21 June 2017 Received in revised form 10 August 2017 Accepted 8 September 2017

Keywords: Electro-hydraulic load simulator Force tracking Improved GMS friction model Finite-time adaptive sliding mode control

a b s t r a c t This paper addresses the force tracking problem of electro-hydraulic load simulator under the influence of nonlinear friction and uncertain disturbance. A nonlinear system model combined with the improved generalized Maxwell-slip (GMS) friction model is firstly derived to describe the characteristics of load simulator system more accurately. Then, by using particle swarm optimization (PSO) algorithm combined with the system hysteresis characteristic analysis, the GMS friction parameters are identified. To compensate for nonlinear friction and uncertain disturbance, a finite-time adaptive sliding mode control method is proposed based on the accurate system model. This controller has the ability to ensure that the system state moves along the nonlinear sliding surface to steady state in a short time as well as good dynamic properties under the influence of parametric uncertainties and disturbance, which further improves the force loading accuracy and rapidity. At the end of this work, simulation and experimental results are employed to demonstrate the effectiveness of the proposed sliding mode control strategy. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Electro-hydraulic load simulator is a kind of hardware-in-the-loop equipment, which can reproduce the complex load characteristics caused by rocket structural stiffness, rocket nozzle thrust and other factors in rocket launch process. It has been widely used in the development and performance test for the thrust vector servo mechanism of rocket system [1– 3], and directly affects the reliability of rocket launch. Therefore, it is of great significance to develop the electrohydraulic load simulator with high precision and quick response in order to further improve the dynamic performance of rocket servo mechanism. Electro-hydraulic load simulator is generally applied to implement the real-time force loading according to the desired signal. Its loading accuracy is mainly influenced by external disturbances and internal nonlinear friction. Therefore, how to eliminate these influences and achieve a better force tracking performance becomes the research hotspot in this field. To reject external disturbance, the structure of load simulator and force tracking control algorithm have both been studied and improved. Based on the idea of motion synchro-compensation, an accessional hydraulic motor is applied to ensure the synchronization between loading motor and aircraft steering motor for reducing disturbance torque [4,5]. A dual-valve parallel connected structure including high-response flow servo valve and P-Q servo valve is designed to eliminate superfluous force for marine load simulator in Ref. [6]. A structure invariance compensation (SIC) method based on feedforward theory is ⇑ Corresponding author at: College of Mechanical Electronic and Control Engineering, Beijing JiaoTong University, Beijing 100044, China. E-mail address: [email protected] (H. Yan). https://doi.org/10.1016/j.ymssp.2017.09.009 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.

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designed to compensate the extraneous force in Ref. [7] and has been recognized as the most widely used method for load simulator to suppress motion disturbance in industry. Then the quantitative feedback theory (QFT) [8,9] and fuzzy control theory [10] are both introduced to design the force controller for aerodynamic load simulator in order to reproduce the wide spectrum of aerodynamic hinge moment variation for aircraft actuation system test. Considering the external disturbance as a lumped uncertain factor, Guan et al. proposed an integral-type adaptive sliding mode controller for the typical electro-hydraulic system with uncertain items and improved its tracking performance [11]. Also, a nonlinear robust control method is adopted to compensate for the uncertain external disturbances and discontinuous hydraulic properties, which guarantees the transient performance and final tracking accuracy [12,13]. Moreover, impedance control with an observerbased compensation for disturbance [14] and various hybrid controllers are also investigated for the precise force or position control in typical hydraulic servo systems. Truong et al. [15] discussed a grey prediction compensator combined with fuzzy PID force controller to reduce the impact of external disturbance and increase the robustness of hydraulic loading system. Another hybrid algorithm combined improved particle swarm optimization (PSO) algorithm with standard PID control is presented in order to enhance the PID gains’ searching efficiency and help achieve the high precision position control for the nonlinear hydraulic system [16]. On the other hand, frictional nonlinearity may cause large steady-state errors, low speed creeping or piston rod jamming failure in the situation of low speed, large load and poor contact surface lubrication, which severely restricts system performance. To compensate for the effects of nonlinear friction, numerous studies have been carried out based on the classical friction model. Alleyne et al. investigated a Lyapunov-based adaptive control algorithm to compensate for the Stribeck friction and hydraulic parametric uncertainties [17]. Aiming at compensating for the change of friction parameters caused by temperature, lubrication, etc., Zhou et al. established a nonlinear observer for estimating the load disturbance and LuGre friction [18]. Lu et al. developed an improved LuGre model which can describe the smooth switch between motor’s high and low-speed mode, and also designed an adaptive friction compensator [19]. Another improved nonlinear friction description based on the LuGre model is estimated using the adaptive fuzzy technique in Ref. [20], which is compensated by an additional feedforward term under PD control. Then, Yao et al. proposed an improved LuGre friction model with changeable maximum dynamic and static friction torque, and also designed a compensator based on structure invariance principle [21]. Parameters in LuGre model can also be identified by a modified GSO algorithm, and then compensated by a nonlinear sliding mode method to suppress parametric uncertainties [22]. Moreover, to avoid the frictional state jump when the system is stationary or operates at small motion, a smooth switch friction model based on hysteresis and Stribeck characteristic is also derived and able to capture the force at near zero velocity [23]. Most of the above-mentioned research on friction identification and compensation is based on the widely used Stribeck or LuGre model [24]. However, Stribeck model cannot give a description of frictional dynamic characteristics, while LuGre model has a limitation on describing frictional hysteresis in slipping stage. Moreover, the complicated expression of LuGre model always makes the dynamic parameters identification and control design difficult although some improved models and identification methods have been established in previous works, which seriously affects the accuracy and speed of friction identification and lead to the decrease of system control precision. In contrast, the generalized Maxwell-slip (GMS) friction model proposed by Al-Bender et al. in Ref. [25] is able to describe all the friction dynamic characteristics completely, in the meantime, its structure is simple, which is convenient for high precision identification and compensation [26–29]. Based on this, considering the influence of oil temperature change and other factors on friction parameters, an improved GMS model combined adaptive scheme and offline estimation with model uncertainties is proposed in this paper. The advantage of adopting this model is that the nonlinear friction model can be linearly parameterized by introducing adaptive coefficients, which not only helps substantially improve the accuracy of compensation for time-varying friction caused by environmental factors, also ensures much faster adaptive adjustment and convergence of friction parameters than that using online identification methods independently in current literature. For the GMS friction identification, PSO algorithm is adopted in this paper. As a global optimization algorithm, PSO is practical and effective for nonlinear system identification [30,31]. It also has much faster convergence speed and less complicated structure compared with the traditional genetic algorithm (GA) and ant colony optimization algorithm (ACO), moreover, its global convergence can be guaranteed by introducing crossover and mutation factor based on the idea of GA [32,33], which is totally accord with the requirements of rapidity and accuracy for nonlinear friction identification and compensation in high-precision load simulator. Meanwhile, many conventional and intelligent control algorithms are proposed to reject the external disturbances in above-mentioned works, however, hybrid controllers including conventional algorithms such as PID are not based on the system model, which have less effect on suppressing nonlinear disturbance and model uncertainties, while the intelligent control algorithms such as fuzzy control may have difficulty in obtaining fuzzy rules, also the computing process of complicated intelligent control algorithms may be very time-consuming and lead to the hardware implementation failure. Therefore, to achieve the requirements of quick response speed and accuracy for the electro-hydraulic simulator in real-time control, designing an effective finite-time control algorithm with strong robustness is necessary. In recent years, terminal sliding mode control has demonstrated impressive finite-time control performance in various fields such as robot manipulator system, flight control system and so on. Its significant characteristics including better robustness against external disturbances and faster response with finite-time convergence of the tracking errors make it more suitable for the typical electro-hydraulic system with highly nonlinear characteristics and quick response requirement than other intelligent control methods and also linear sliding mode control [22,34], however, the chattering reduction is considered as an important factor during the control design due to the discontinuous state feedback. Ullah et al. [35] presented a fractional order adaptive

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fuzzy continuous nonsingular terminal sliding mode controller to estimate the extra disturbance for the hydraulic system. Then, based on the predefined nonsingular terminal sliding manifold and the Lyapunov stability theory, a class of absolutely continuous robust controllers is discussed for robotic manipulators to counteract both uncertain dynamics and unbounded disturbances [36]. In Ref. [37], system uncertainties and unknown disturbances are approximated by an adaptive neural network in order to decrease the chattering, and then a fraction integral terminal sliding mode controller is designed for inchworm robot manipulator to develop performance. Besides, by using bat algorithm to tune the controller parameters, more hybrid controllers based on the integral sliding mode control are proposed to apply in the different kinds of robot manipulator systems [38,39]. Based on the above analysis, we designed an improved finite-time adaptive terminal sliding mode controller in order to compensate for the nonlinear GMS friction more accurately and improve the system robustness effectively. The nonlinear sliding surface is designed to be continuous and the derived control law is simpler compared with previous methods, which are proved effective for chattering and tracking error reduction and convenient for hardware implementation. Also, the boundaries of chattering and force tracking error can be adjusted by specific control parameters when taking into account the system dynamic performance. In particular, the finite-time control scheme is introduced into the electrohydraulic servo system for the first time, and the finite-time characteristic including specific system convergence time is calculated and analyzed, which demonstrates both better force control performance under the influence of nonlinear effects and faster response than conventional methods. Overall, the main contributions of this paper are listed as follows: (1) A nonlinear model of electro-hydraulic load simulator including improved GMS friction model is established to increase the system model accuracy. (2) The PSO identification algorithm is employed and the system hysteresis is also analyzed to identify the accurate GMS friction model. (3) A finite-time nonlinear adaptive sliding mode controller is designed, and the boundary of force tracking error and convergence time are derived to guarantee the system performance. To test the force tracking performance by using the proposed controller, simulation and experiment results are both obtained. 2. Problem formulation and dynamic model 2.1. Dynamic model of force loading system The electro-hydraulic load simulator is composed of position servo device, force loading device, friction loading device, elastic loading device and inertia regulating device, shown in Fig. 1. The position servo device and force loading device are both closed-loop system, which comprises servo valve and double-rod cylinder with the piston rod hinged at the end of roof beam. The friction loading device, elastic loading device and inertia regulating device are all installed on the shaft of roof beam used to adjust the equivalent viscous damping coefficient, spring constant and moment of system inertia. The force loading device is a typical electro-hydraulic force servo system coupled with disturbance generated by position servo device. The control voltage of servo valve and the driving force of cylinder are regarded as its input and output,

Fig. 1. Structure of the electro-hydraulic load simulator.

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respectively, and the goal is to have the loading cylinder’s driving force track any desired force signal with minimum error in the shortest time. Then the loading force is transferred through the beam to act on the position servo device. Taking the force servo system as an object, its basic work principle is illustrated in Fig. 2. The flow equation of servo valve in force servo system can be written as follows [40]

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q L ¼ C d wxv ðps  sgnðxv ÞpL Þ

q

ð1Þ

where Q L is the load flow, C d is the discharge coefficient, w is the spool valve area gradient, xv is the servo valve spool displacement, sgnðxv Þ is the sign function depending on whether the valve is open in positive or negative direction, q is the oil density, ps is the supply pressure, pL is the load pressure. For a double-rod cylinder hydraulic system, the original control volumes of two chambers in loading cylinder are assumed to be the same, also the external leakage of the cylinder is neglected here. Therefore, the dynamics of cylinder oil flow can be written as follows

Q L ¼ Ac

dx V t dpL þ þ C tc pL dt 4be dt

ð2Þ

where x is the displacement of the force loading piston rod, Ac is the ram area of the two chambers, V t is the total control volume including cylinder and pipelines of the servo valve, be is the effective bulk modulus of the hydraulic fluid in the container, C tc is the coefficient of internal leakage of the cylinder. Taking vertical downward direction along the x axis as positive direction, the dynamics of piston rod’s force balance can be expressed as 2

m0

d x dt

2

¼ Ac pL  B0

dx  ks x  m0 g  F f  f d ðtÞ dt

ð3Þ

where m0 is the equivalent mass of loading cylinder and load, B0 is the equivalent viscous damping coefficient of friction loading device, ks is the equivalent spring constant of the elastic force loading device, F f is the nonlinear friction, f d ðtÞ is the uncertain external disturbance which is generated by position servo device. The following equation is given to describe the relationship between control voltage and spool displacement when the natural frequency of servo valve is far greater than that of the cylinder.

x_ v ¼ xv xv þ K v xv u

ð4Þ

where u is servo valve’s input current, xv is the natural frequency of servo valve, K v is the equivalent gain coefficient of servo valve based on its prestage structure.

Fig. 2. Schematic diagram of the electro-hydraulic load simulator.

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2.2. Improved GMS friction model The total friction existed in force servo system is composed of two parts including friction between the piston rod and cylinder and also friction generated by the rotation of spherical hinge. Due to the much smaller value, the spherical hinge friction can be neglected here, only the component of friction between the piston rod and cylinder is considered. In this paper, GMS friction model is introduced to describe the nonlinear friction characteristics in force loading cylinder. It consists of a number of elastic-slip elementary blocks connected in parallel, which are all excited by the velocity of loading piston rod. Each elementary block represents a pair of convexities with specific hysteresis characteristic and frictional state. The total friction is the sum of all the elementary forces, shown in Fig. 3. Considering the influence of viscous friction, the total friction acting on force loading cylinder can be expressed by

Ff ¼

N X

F if þ rf x_

ð5Þ

i¼1

where N is the number of elementary blocks connected in parallel, F if is the friction generated by each elementary block, rf is the viscous friction coefficient. x_ is the velocity of the force loading piston rod, which is regarded as the input for friction model. To describe different states of the specific elementary block more precisely, each elementary force F if can be expressed as a piecewise function. (1) At the reversal point of spool, x_ tends to zero, therefore, the elementary block is sticking which contains hysteresis and pre-sliding state, the equation is given by i

dF f ¼ ki x_ dt

ð6Þ

where ki is the stiffness coefficient of the specific elementary block. Define the maximum friction of each elementary block in P _ where ai is the weight coefficient of each elementary block and satisfies Ni¼1 ai ¼ 1, sðxÞ _ represents sticking state as ai sðxÞ, the Stribeck function, which is generally expressed as follows x_

_ ¼ F c þ ðF s  F c ÞeðjV s jÞ sðxÞ

d

ð7Þ

where F c is the Coulomb friction, F s is the static friction, V s is the Stribeck velocity and d is the shaping factor. _ is satisfied, the state of elementary block changes to slipping which represents fric(2) When the inequality F if P ai sðxÞ tional lag and Stribeck effect. The equation is given by

! i dF f Fi _ ai  f ¼ C sgnðxÞ _ dt sðxÞ

ð8Þ

where C is the constant term used to describe the rate that slipping friction is convergent to the Stribeck curve. Considering the piston rod slips at a constant velocity, the Eq. (8) can be converted into

_  ai sðxÞ _ F isf ¼ sgnðxÞ

ð9Þ

where F isf is the steady-state slipping friction of specific elementary block. Define the dynamic effect of slipping state as did , therefore the slipping friction of each block is obtained as

_  ai sðxÞ _ þ did F if ¼ F isf þ did ¼ sgnðxÞ

ð10Þ

In order to give a complete description of the friction states including sticking and slipping for each elementary block, a state factor c is introduced



ci ¼

1 if  is true 0

otherwise

;

ði ¼ 1; 2; . . . ; NÞ

ð11Þ

The explanation for Eq. (11) is that equations cstick ¼ 1 and cslip ¼ 0 are satisfied when block i is sticking, otherwise cslip ¼1 i i i ¼ 0. From Eqs. (11), the general expression of friction for each elementary block is expressed as and cstick i

_ _ F if ¼ cstick ki xðtÞ þ cslip i i ðsgnðxÞ  ai sðxÞ þ did Þ

ð12Þ

Therefore, substituting Eq. (12) into Eq. (5), the Eq. (5) is converted into

F f ¼ AT B þ rf x_ þ dD

ð13Þ

_ a2 sðxÞ; _ . . . ; aN sðxÞ _ is the parameter of the GMS friction model which needs to be identified by where A ¼ ½k1 ; k2 ; . . . ; kN ; a1 sðxÞ; h i slip stick stick stick _ slip _ _ experiment, B ¼ c1 xðtÞ; c2 xðtÞ; . . . ; cN xðtÞ; cslip represents friction state variable, 1 sgnðxÞ; c2 sgnðxÞ; . . . ; cN sgnðxÞ PN slip dD ¼ i¼1 ci did is the total slipping dynamic term which acts as a perturbation.

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Fig. 3. Schematic diagram of GMS friction model.

_ denote the estimated value of Stribeck effect obtained by experimental identification and ds denote the estimated Let ^sðxÞ error between identified model and true friction, therefore, we have

^sðxÞ _ ¼ sðxÞ _  ds

ð14Þ

Assumption 1. For any bounded velocity interval Xx_ ¼ ½x_ min ; x_ max , the supremum of ds exists and can be expressed as

supjds j 6 Ds Xx_

ð15Þ

where Ds is defined as the boundary of estimated error for Stribeck model. Considering the dynamic items in slipping state, the estimated error of identification and also the parametric uncertainties in GMS model, substituting Eq. (14) into Eq. (13), thus the improved expression of GMS friction can be converted into

^T B ^ þ br ^ f x_ þ D F f ¼ aA

ð16Þ

where a and b are the adaptive coefficients used to compensate for the parametric uncertainties in GMS model caused by the change of system temperature or lubrication state, etc. during the work process, and satisfy a > 0, b > 0. P _ D ¼ Ni¼1 cslip i ðsgnðxÞ  ai ds þ did Þ is the adaptive error term including modeling and estimated errors used to further improve the accuracy of nonlinear friction description. Ideally, when modeling and identification are accurate enough, the following ^ f is the estimated viscous friction coefficient, and other estimated variables are equations are given by a ¼ b ¼ 1 and D ¼ 0. r defined as

h i ^¼ c ^stick ^stick ^slip ^slip ^slip ^stick xðtÞ; c A 1 2 xðtÞ; . . . ; cN xðtÞ; c1 sgnðv Þ; c2 sgnðv Þ; . . . ; cN sgnðv Þ ; h i ^2 ; . . . ; k ^N ; a ^1 ; k ^¼ k ^ 1^sðxÞ; ^ 2^sðxÞ; ^ N^sðxÞ _ a _ ...;a _ : B

^ B ^ in GMS model can be obtained by offline PSO identification which is stated in ^ f ; A; The estimated friction parameters r Section 3. Assumption 2. The structure parameters of electro-hydraulic load simulator are bounded. Based on Assumption 1 and 2, the boundedness of total modeling error in GMS friction model can be proved, which is stated in Lemma 1. Lemma 1. For the force servo system (1)–(4) with bounded structure parameters, the total error term D in Eq. (16) is bounded. Proof. Define a Lyapunov function for each elementary block as

V if ¼

1 i 2 ðF Þ 2 f

Then the derivative of Eq. (17) along Eq. (8) can be expressed as

ð17Þ

S. Kang et al. / Mechanical Systems and Signal Processing 102 (2018) 117–138

_ ai  V_ if ¼ F if F_ if ¼ CjF if jsgnðF if ÞsgnðxÞ

! jF if j _ sðxÞ

123

ð18Þ

When a block is in the slipping state, the sign of F if and x_ are always different from zero, therefore, and V_ if is negative definite if

jF if j 6 jF isf j < ai F s

ð19Þ

From Eqs. (10) and (19), since F if and F isf are bounded, the dynamic item did must be bounded. And the estimated error ds is also bounded based on Assumption 1. Therefore, the conclusion can be obtained that the total error term D in GMS model is bounded. The advantage of using the improved GMS model (16) combined adaptive items with offline estimated parameters is that the nonlinear friction model can be linearly parameterized by introducing adaptive coefficients, which not only helps improve model accuracy for compensating the uncertainties caused by environmental factors, also provides the convenience for adaptive laws design and ensures much faster system adaptive response than the parameter adjustment by using online identification method. 3. Identification of GMS friction model In this section, the experimental identification for GMS friction model is discussed, which can be divided into two steps: static parameters identification and dynamic parameters identification. In the case of ensuring the friction identification accuracy, PSO algorithm is more suitable for solving the optimization problem of continuous function such as Stribeck function than GA and ACO when taking into account the convergence speed. At the same time, considering the higher precision but longer time of PSO online identification for complex function than direct adaptive adjustment, to achieve the requirement of system rapidity and accuracy for real-time implementation, PSO offline identification is adopted in this section, and the aim is to obtain the basic parameterized GMS model which can guarantee the model accuracy and also provide convenience for the following direct adaptive law design in Section 4.2 in order to improve the efficiency of parameter adaptive adjustment. 3.1. Static parameters identification When the piston rod is placed horizontally and slips at a constant velocity in the limitation of stroke with no position disturbance, from Eqs. (3), (5), (7), and (9), the force balance equation of piston rod can be expressed as follows

  x_ d _ F c þ ðF s  F c ÞeðjV s jÞ F driv e ¼ Ac pL ¼ ðB0 þ rf Þx_ þ ks x þ sgnðxÞ

ð20Þ

^s ^ ^ s ¼ ½F^c F^s V ^ f , the reference d r According to Eq. (20), define the static friction parameters need be identified as P signals are enforced by a PID controller in order to drive the piston rod move at the constant velocities of [±2, ±3, ±4, ±6, ±8, ±10, ±12, ±14, ±16, ±20, ±25, ±30, ±45, ±60, ±75, ±90, ±105, ±120, ±135, ±150] mm/s. At the same time, the displacement of the piston rod and driving force of cylinder, regarded as the input and output for identification, are measured by displacement sensor and force sensor, respectively. Then the data is collected in six independent experiments and substituted into Eq. (20) to search the global optimum solution for Ps by using the efficient PSO algorithm which minimizes the fitness function as follows

J¼

n  2 1X F kdriv e  F^kdriv e 2 k¼1

ð21Þ

where F kdriv e and F^kdriv e are the measured and estimated driving force of cylinder in each experiment group respectively, n is the total number of the test group, n ¼ 40. ^ s ¼ 2:031 mm=s, ^ According to Eq. (21), the identified Stribeck parameters are F^c ¼ 640:8 N, F^s ¼ 865:4 N, V d ¼ 0:8764,

r^ f ¼ 7:265 Ns=mm, and the computation time is 1.613 s. Fig. 4 shows the experimental data and identified Stribeck curve.

Then the errors between fitted parameters and experimental values are obtained as eðF c Þ ¼ 3:45%, eðF s Þ ¼ 2:26%, eðV s Þ ¼ 1:89%, eðdÞ ¼ 3:62% and eðrf Þ ¼ 1:37%. For comparison, GA is also adopted in the experiment for Stribeck function identification, then we can obtain the errors between fitted parameters and experimental values as eðF c Þ ¼ 2:94%, eðF s Þ ¼ 3:18%, eðV s Þ ¼ 1:04%, eðdÞ ¼ 4:05%, eðrf Þ ¼ 0:96%, and the computation time is 2.074 s. In conclusion, using the less complicated PSO for friction identification can achieve the similar accuracy as using GA, and it also has a more stable trend of extreme variation and needs shorter computation time, which proves the practicability and effectiveness of PSO. It is also worth noting that the parametric uncertainties in friction are caused by several factors, such as oil temperature and sealing condition etc., resulting from the continuous operation.

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Fig. 4. Measured and identified Stribeck curve.

3.2. Dynamic parameters identification To identify the stiffness coefficient ki and weight coefficient ai for each elementary block, as proposed in Ref. [27], a sinusoidal signal with small amplitude is enforced to drive piston rod move as a cyclic motion in the displacement range of [1.2, 1.2] mm in order to simulate the transition from pre-sliding to sliding. Thus, the hysteresis characteristic of loading cylinder is illustrated in Fig. 5. Define the number of elementary block as N ¼ 3, therefore, the ascent trajectory from the initial translational state to the reversal state in hysteresis curve can be approximated by a piecewise linear function including three segments. For the next identification, the segmented nodes N 0 -N 3 of piecewise function are selected manually, which are also shown in Fig. 5. Where N i represents each state during pre-sliding and K i denotes the slope of each segment. Thus, the corresponding stiff^i ði ¼ 1; 2; 3Þ for each elementary block can be calculated as follows according to Eq. (5) ness coefficient k

^ ¼K K k i i iþ1

ð22Þ

Fig. 5. Cylinder hysteresis curve in pre-sliding state.

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where K 4 ¼ 0. Then considering Eqs. (5) and (6) and the constraint equation can be determined as follows

a^ i ¼

3 X 1 ^ xN  xN  k F N i  F N0  j 0 i ^sðxÞ _

PN i¼1

ai ¼ 1, the weight coefficient a^ i ði ¼ 1; 2; 3Þ

!

ð23Þ

j¼iþ1

where j denotes the elementary block which is still sticking, F Ni and xNi denote the force and position at state N i , respectively. The identified GMS dynamic parameters are shown in Table 1 The convergent rate C = 1650 N/s is selected to minimize the error between identification result and actual displacement, ^i can be obtained according to the cylinder velocity and Eqs. (6) and (7) with the estimated model thus the state factors c parameters above. 4. Finite-time adaptive sliding control design 4.1. Design model and issues to be addressed Due to the proportional relationship between load pressure and loading force, the force control problem can be equivalent T to pressure control for the following analysis. Therefore, define the state variables as x ¼ ½x1 x2 x3 x4 T ¼ ½x x_ pL xv  and the output variable y ¼ Ac pL , then the entire system including Eqs. (1)–(4) can be expressed in state-space form as follows

8 x_ 1 ¼ x2 > > > > > x_ 2 ¼ m10 ðks x1  B0 x2 þ Ac x3  m0 g  F f  f d ðtÞÞ > > <  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x_ 3 ¼  4AVctbe x2  4CVtctbe x3 þ 4CV tdpbeffiffiqffiw ps  sgnðx4 Þx3 x4 > > > > > x_ 4 ¼ xv x4 þ K v xv u > > : y ¼ Ac x3

ð24Þ

From Eq. (24), it is shown that the force servo system is a nonlinear minimum phase system with external disturbances. For ease of controller design, define the parameters set e ¼ ½e1 e2 e3 e4 e5 T as e1 ¼ 4Ac be =V t , pffiffiffiffi e3 ¼ 4C d be w=ðV t qÞ, e4 ¼ xv , e5 ¼ K v xv , thus the state space Eq. (24) along Eq. (16) is transformed to

8 x_ 1 ¼ x2 > > h i   > > > 1 ^T ^ ^ _ > > < x2 ¼ m0 ks x1  B0 x2 þ Ac x3  m0 g  aA B þ brf x2 þ D  f d ðtÞ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x_ 3 ¼ e1 x2 þ e2 x3 þ e3 ps  sgnðx4 Þx3 x4 > > > > > x_ 4 ¼ e4 x4 þ e5 u > > : y ¼ Ac x3

e2 ¼ 4C tc be =V t ,

ð25Þ

Therefore, the parametric uncertainties in nonlinear friction model and external disturbance item are combined with the nonlinear hydraulic characteristic parameters set e. Based on the realistic system, the following assumption is given. Assumption 3. The desired force function F desire is two-order continuous and differentiable and bounded, therefore, the loading cylinder’s position, velocity and acceleration are all bounded. Based on the practical situation, the disturbing force f d ðtÞ is assumed to be continuous and bounded, which can be expressed as sup jf d ðtÞj 6 Df , where Df is defined as the boundary of external disturbance.

4.2. Finite-time adaptive sliding mode control law design Since the force loading system contains bounded external disturbance and uncertain GMS friction with state switch, a finite-time adaptive sliding mode method is adopted to eliminate the impact on system accuracy and ensure a quick response. First, define the loading pressure error as follows

Table 1 Identification results of dynamic parameters in GMS model. Parameter

^ (N/m) k a^

Block 1

2

3

1884

196.9

184.1

0.829

0.089

0.082

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e3 ¼ x3  x3d

ð26Þ

where x3d is the desired loading pressure. In order to ensure the good dynamic performance and high steady accuracy of load simulator system, a continuous nonlinear sliding surface based on sigmoid function is designed as

s ¼ ne_ 3 þ /ðk; g; e3 Þ

ð27Þ

where n, k and g are tuning parameters, and satisfy n > 0; k > 0; g > 0, /ðe3 Þ ¼ k=½1 þ expðge3 Þ  k=2. When the system state arrives at s surface, we have

e_ 3 ¼ 

 k 1 1   g e n 1þe 3 2

ð28Þ

From Eq. (28), it is known that pressure error e3 will approach zero at a large constant velocity when system initial state is far away from equilibrium point so as to ensure the rapid dynamic response and finite convergence time, and also have a rather high sensitivity when its sliding state changes in a small neighborhood around equilibrium point, which guarantees system’s steady accuracy. Define the Lyapunov function as

V1 ¼

1 m0 s2 2

ð29Þ

To simplify the following analysis, define g 1 ¼ ks =m0 x1  B0 =m0 x2 þ Ac =m0 x3  g, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 3 ¼ e3 ps  sgnðx4 Þx3 , then differentiating Eq. (29) along Eq. (25) yields

g 2 ¼ e1 x2 þ e2 x3

h  i  ^T B ^ þ br ^ f x2 þ D þ ne1 f d þ m0 ne5 g 3 u þ w V_ 1 ¼ m0 ss_ ¼ s ne1 aA

where

and

ð30Þ



 @g @g 1 @/ w ¼ m0 n e1 g 1 þ e2 g 2 þ e2 g 3 þ g 2 3 þ g 3 3 x4 þ e4 g 3 x4  €x3d þ ðg 2 þ g 3 x4  x_ 3d Þ : n @e3 @x3 @x3

For system (25) with uncertain parameters a; b and D, the adaptive control method is introduced and combined with sliding mode control to compensate for the frictional uncertainties. ^ , respectively, then the estimated errors are obtained as ^ and D ^; b Define the estimates of a; b and D as a

~ ¼DD ^ ^ D a~ ¼ a  a^ ; b~ ¼ b  b;

ð31Þ

In order to derive the adaptive laws, the following Lyapunov function is defined

V2 ¼ V1 þ

1 2 1 ~2 1 ~2 b þ a~ þ D 2u 1 2u2 2u3

ð32Þ

where u1 ; u2 and u3 are positive definite constants. Taking into account Eqs. (32) and (30), the time derivative of V 2 is

    1 _ 1 ^T B ^ þ ne1 b ^ þ ne1 f þ w þ m0 ne5 g su þ a ^r ^A ^s ~ ne1 ^hT x ^ f x2 þ ne1 D V_ 2 ¼ s ne1 a a^ þ b~ ne1 r^ f x2 s  b^_ d 3 u1 u2  1 ^_ ~ ne1 s  þD D

u3

ð33Þ

Thus the controller can be designed as

u¼

u1 þ u2 þ u3 m0 ne5 g 3

ð34Þ

^T B ^  ne1 b ^ is the adaptive controller which is used to compensate parametric uncertainties in ^r ^A ^ f x2  ne1 D where u1 ¼ ne1 a friction, and u2 ¼ w is used to compensate for the certain variables in the system. To eliminate the chattering phenomena caused by sliding mode control and further decrease the force loading steady error, u3 is designed as a robustifying item to reject the bounded external disturbance f d ðtÞ. Considering the characteristic of sliding surface, the robust controller is designed as follows based on Lemma 1 and Assumption 3

u3 ¼ hs  hne1 Df

s b

ð35Þ

where h and b are both positive constants. The adaptive laws for improving the accuracy of identified GMS model and rate of parameter adaptive adjustment are chosen as

^ a^_ ¼ u1 ne1 A^ T Bs

ð36Þ

^_ ¼ u ne1 r ^ f x2 s b 2

ð37Þ

S. Kang et al. / Mechanical Systems and Signal Processing 102 (2018) 117–138

^_ ¼ u ne1 s D 3

127

ð38Þ

4.3. Analysis of system stability and finite-time convergence Theorem 1. For the electro-hydraulic force servo system (25) consisting of nonlinear friction (16) with uncertain parameters and bounded external disturbances, satisfying Assumption 1–3, under the controller (34) and adaptive laws (36)–(38), its force tracking error ey will converge to a small neighborhood of equilibrium point in finite time, and the system is asymptotically stable. Proof. Substituting controller Eq. (34) and the friction parameters adaptive laws (36)–(38) into Eq. (33) yields

s2 2 V_ 2 ¼ hs þ ne1 f d s  hne1 Df b

ð39Þ

Based on Assumption 3, Eq. (39) satisfies the following inequality when jsj P b=h

 jsj2 jsj 2 2 V_ 2 6 ne1 jf d jjsj  hs  hne1 Df ¼ hs  ne1 hDf  jf d j jsj 6 0 b b

ð40Þ

which means that jsj will always converge to be within the boundary ½b=h; b=h in finite time and remains inside thereafter. Inside the boundary, the system state will slide along Eq. (27) to the equilibrium point. To determine the boundary of force tracking error, define another Lyapunov function as follows

V3 ¼

1 2 e 2 3

ð41Þ

Then the time derivative of V 3 along Eq. (27) is obtained

e3 e3 e3 b V_ 3 ¼  /ðe3 Þ þ s 6  /ðe3 Þ þ je3 j nh n n n

ð42Þ

When pressure error e3 satisfies inequality as follow

je3 j P 

1

g

 ln 1 

4b kh þ 2b



ð43Þ

V_ 3 is negative semi-definite, thus the pressure error e3 will converge to the boundary in finite time, that is

limje3 j 6 

t!T s

1

g

 ln 1 

4b kh þ 2b



ð44Þ

n   o 4b Therefore, the system state trajectory will reach the set X ¼ jey j 6  Ag ln 1  khþ2b ; jsj 6 hb in finite time, and the system is asymptotically stable. h The total convergence time that system varies from the initial state to the steady state can be determined by Eqs. (27) and (44) as

T s 6 t0 þ

 2n 4n  ln 1 þ ege3 ðt0 Þ  e3 ðt0 Þ kg k

ð45Þ

where t0 represents the time that system state arrives at the boundary of attraction region from its initial state, e3 ðt0 Þ denotes the pressure error at t0. Due to the converge rate based on exponential approach law during the reaching segment, t0 can be estimated as follows

t0 6



  1 b b  ln ln sð0Þ þ h h hne1 Df

ð46Þ

where s(0) is the system initial state. Based on Eq. (45) and (46), we can conclude that the system tracking error will converge to the small neighborhood of zero in finite time. Remark 1. Eqs. (44)–(46) indicate that the force tracking accuracy and convergence time are both related to the dynamic parameters ðk; gÞ in the sliding surface and robust parameters ðh; bÞ. k is position coefficient, which determines the convergence speed and steady state precision of force tracking error. g is exponential coefficient, which mainly affects the steady state accuracy of the system. h denotes the amplitude of robust control law, which determines the thickness of attraction region for stable sliding with b. It is worth noting that the force tracking error and convergence time will reduce when increasing k; g; h and decreasing b, but at the same time, system dynamic performance will be affected by the chatting phenomena and narrow attraction region, etc. Therefore, it is important to obtain the tradeoff when selecting control parameters.

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5. Simulation and experimental results 5.1. Simulation results To illustrate the validity of above design, first, a structure invariance compensator (SIC) without friction compensation is adopted to provide for comparison, then the effectiveness and rapidity of proposed finite-time adaptive sliding mode controller (FASMC) are demonstrated by examining the force tracking performance under the influence of disturbances with different amplitudes and frequencies. The parameters of practical electro-hydraulic system are shown in Table 2. In the simulation, two kinds of reference force signal given as F desired ¼ 10 ðkNÞ and F desire ¼ 10 sinðptÞ (kN) are firstly designed to verify the performance of proposed FASMC, and then more reference signals including triangular signal and irregular continuous signal are also provided to demonstrate the effectiveness of improved GMS model based friction compensation on increasing force control accuracy compared with the conventional Stribeck friction compensation. The sampling period is set to 1 ms, which is the same as that in the hardware-in-the-loop test. The simulation results for these two cases are presented as follows: Case A: Structure invariance compensator (SIC) without friction compensation SIC method has been applied by most of the industrial load simulator to suppress external disturbance. In the simulation, the disturbance is firstly defined as f d1 ¼ 2 sinð0:5ptÞ (kN) according to the Assumption 3, thus the force tracking trajectories and corresponding tracking errors for two desired force signals are shown in Figs. 6 and 7, respectively. Then the disturbance is changed to f d2 ¼ 5 sinð2ptÞ (kN), and the force tracking performance for two different reference signals are shown in Figs. 8 and 9, respectively. By comparison, the loading system has a smaller amplitude of stable tracking error (average 112 N) and shorter convergence time (1.2 s) when inputting a constant signal, but larger chattering in the steady-state process due to static friction. When inputting a sinusoidal signal, system stabilizes in 2.2 s, and the maximum stable tracking error is about 276 N. As shown in Figs. 6–9, the amplitude of force tracking error, overshoot and convergence time all rise with the increase of disturbance’s amplitude and frequency, no matter the input reference signal is constant or sinusoidal. From Figs. 8 and 9, influenced by a larger external disturbance, the amplitude of stable tracking error rises to 223 N and also the convergence time increases to 1.6 s when inputting a constant signal, while system stabilizes in a longer time (more than 6 s) with an extreme high overshoot and its maximum stable tracking error reaches about 392 N when inputting a sinusoidal signal. Also, the severe chattering phenomenon shown in Figs. 6–9 has not been eliminated due to the absence of friction compensation, which contributes to the decrease of control accuracy and the damages on simulator equipment. The analysis results concluded that effective friction compensation is the key factor to further improve system accuracy in high-precision load simulator. Case B: Finite-time adaptive sliding mode controller (FASMC) The effectiveness of the proposed FASMC method is also illustrated for the two desired force signals mentioned above. The controller parameters and initial conditions are set as follows: the sliding surface parameters are set as n ¼ 1; k ¼ 12; g ¼ 60 considering both the tracking error convergence rate and system stability, the adaptive rates are chosen as u1 ¼ 103 ; u2 ¼ 450; u3 ¼ 800, and the robust parameters are set to h ¼ 25; b ¼ 107:5 for balancing the upper bound of tracking error and reaching performance, then the upper bound of uncertain external disturbance term is defined as ^ are chosen as ^ D ^ ; b; Df ¼ 5:5kN according to practical situation, and the initial values of estimate a ^ ð0Þ ¼ 0:01. The GMS friction parameters are set as the identification results in Table 1. ^ a^ ð0Þ ¼ bð0Þ ¼ 1; D

Two control schemes are provided for comparison: (I) FASMC with Stribeck friction compensation; (II) FASMC with improved GMS friction compensation. When the disturbance is chosen as f d1 ¼ 2 sinð0:5ptÞ (kN), the force tracking trajectories and tracking errors for two desired force signals under control scheme I and II are shown in Figs. 10 and 11, respectively.

Table 2 Parameters of electro-hydraulic load simulator. Parameters

Value

Parameters

Value

ps

7 MPa 840 kg/m3 6.9  108 Pa 0.62 0.0267 m 4.7  1012 m5/N s 1.5  103 m3

m0 Ac L k0 B0

70 kg 2.6  103 m2 0.5 m 4  104 N/m 60 17.74 Hz 1

q be Cd w Ctc Vt

xv Kv

S. Kang et al. / Mechanical Systems and Signal Processing 102 (2018) 117–138

Fig. 6. Simulation of force tracking performance under SIC compensation. With a constant reference signal and small disturbance.

Fig. 7. Simulation of force tracking performance under SIC compensation with a sinusoidal reference signal and small disturbance.

Fig. 8. Simulation of force tracking performance under SIC compensation with a constant reference signal and large disturbance.

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Fig. 9. Simulation of force tracking performance under SIC compensation with a sinusoidal reference signal and large disturbance.

Fig. 10. Simulation of force tracking performance under FASMC controller with a constant reference signal and small disturbance.

When the disturbance is changed to f d2 ¼ 5 sinð2ptÞ (kN), the force tracking performance under different reference signals and control schemes are shown in Figs. 12 and 13, respectively. As shown in Figs. 10–13, when adopting proposed FASMC algorithm, the amplitude of force tracking error, overshoot and convergence time all decrease significantly under different disturbances and control schemes. From Figs. 10 and 11, when imposing a small disturbance, the system tracking errors can keep in a relatively smaller ranges of [4.93, 4.93] N and [37.8, 37.8] N under scheme II than [18.1, 18.1] N and [95.4, 95.4] N under scheme I with different reference signals, and the convergence time under two control schemes both reduce to 0.8 s and 1.8 s with a slight overshoot compared to case A. Also, with the larger disturbance shown in Figs. 12 and 13, the system tracking errors keep in the ranges of [157, 157] N and [192, 192] N under scheme I, [42.1, 42.1] N and [91.3, 91.3] N under scheme II, respectively. And the convergence time under two control schemes both reduce to 1.3 s and 3.3 s accordingly compared to case A. Due to friction compensation and robust item in the control law, the tracking curves become much smoother, only jumping slightly at the signal reversals under scheme I and totally smooth under scheme II. Moreover, by analyzing the amplitude of tracking error curves under different disturbances, it can be observed that applying improved GMS model based friction compensation can dramatically decrease the force tracking error by average 73.2% when inputting the constant signal and 56.7% when inputting the sinusoidal signal compared with the conventional Stribeck model based friction compensation.

S. Kang et al. / Mechanical Systems and Signal Processing 102 (2018) 117–138

131

Fig. 11. Simulation of force tracking performance under FASMC controller with a sinusoidal reference signal and small disturbance.

Fig. 12. Simulation of force tracking performance under FASMC controller with a constant reference signal and large disturbance.

To further verify the effectiveness of improved GMS model based friction compensation, the triangular wave signal and irregular continuous signal are also introduced as the force reference signal for the simulation verification. When the disturbance is set as fd1, the force tracking trajectories and tracking errors for these two desired force signals under control scheme I and II are shown in Figs. 14 and 15, respectively. Then, the disturbance is increased to fd2, and the force tracking performance under different reference signals and control schemes are shown in Figs. 16 and 17, respectively. From the simulation results illustrated in the Figs. 14–17, the decrease of force tracking errors and convergence time can be observed. Compared Figs. 14 and 16, when imposing a triangular wave reference signal with different disturbances, system tracking errors can keep in the ranges of [87.6, 87.6] N and [176, 176] N under control scheme I, which is much wider than the tracking errors’ ranges under scheme II ([34.2, 34.2] N and [79.8, 79.8] N). Also, the convergence time under two control schemes are both reduced to 1.8 s and well satisfy the requirement of system rapidity. Meanwhile, as shown in Figs. 15 and 17, when imposing an irregular reference signal with different disturbances, the system tracking errors stay in the ranges of [75.3, 75.3] N and [197, 108] N under scheme I, [24.7, 24.7] N and [61.6, 37.2] N under scheme II,

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Fig. 13. Simulation of force tracking performance under FASMC controller with a sinusoidal reference signal and large disturbance.

Fig. 14. Simulation of force tracking performance under FASMC controller with a triangular wave signal and small disturbance.

respectively, thus it can be seen that the force tracking effect under the irregular signal is slightly better than that under the triangular wave signal due to the less signal reversal. Moreover, compared the force tracking curves under different friction compensation methods, we can conclude that the tracking errors’ curves under scheme II are much smoother than that under scheme I even at the signal reversals. Also, the amplitude of tracking errors under the control scheme I can be decreased by 61.8% when inputting the triangular wave signal and 68.7% when inputting the irregular signal by adopting the more accurate GMS model based friction compensation, which proves the necessity of accurate modeling for nonlinear friction in order to improve the system performance. Based on the discussion above, the relative error rate of force tracking obtained by applying different control methods and desired signals are summarized in Table 3. The parameter estimations of a; b and D for improved GMS friction model when inputting sinusoidal signal with disturbance fd2 are shown in Fig. 18. It can be seen that all the estimations are bounded in a relatively small range and converge within about 3.3 s, though there exists fluctuation during value updating due to the direct-type adaptive laws.

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133

Fig. 15. Simulation of force tracking performance under FASMC controller with an irregular signal and small disturbance.

Fig. 16. Simulation of force tracking performance under FASMC controller with a triangular wave signal and large disturbance.

From the comparison of simulation results in Table 3, it can be observed that the force loading system under proposed FASMC algorithm with improved GMS friction compensation can always achieve the much smaller tracking error boundaries and shorter convergence time than those adopting conventional SIC algorithm and also Stribeck friction compensation method no matter inputting what kinds of reference signal under the influence of high frequency sinusoidal perturbation, which not only verifies the effectiveness of proposed FASMC algorithm, also demonstrates the feasibility of improved GMS model based friction compensation for high precision electro-hydraulic load simulator. 5.2. Experimental results The experimental test rig is shown in Fig. 19. In the load simulator, the cylinder on the left is force loading cylinder, while the right cylinder is utilized to simulate thrust vector servo mechanism for providing motion disturbance. The specifications of key components adopted in test rig are listed in Table 4. The mechanical structure parameters are shown in Table 2 and the FASMC control parameters are set as given in case B of Section 5.1. The sampling time is set to 1 ms.

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Fig. 17. Simulation of force tracking performance under FASMC controller with an irregular signal and large disturbance.

Table 3 Relative error rate of force tracking under different control methods and desired signals. Algorithm

Signal fd1

SIC FASMC I FASMC II

fd2

Constant (%)

Sine (%)

Triangle (%)

Irregular (%)

Constant (%)

Sine (%)

Triangle (%)

Irregular (%)

1.12 0.181 0.049

2.76 0.954 0.378

– 0.876 0.342

– 0.753 0.247

2.23 1.57 0.421

3.92 1.92 0.913

– 1.76 0.798

– 1.97 0.616

Fig. 18. Parameter estimations with the disturbance fd2.

S. Kang et al. / Mechanical Systems and Signal Processing 102 (2018) 117–138

135

Fig. 19. Electro-hydraulic simulator test rig.

To verify the effectiveness and practicability of proposed FASMC algorithm, the disturbances are set as the same as defined in Section 5.1, which is generated by the displacement of the right cylinder. As the most widely used signals in the performance test of the rocket system, the constant signal and sinusoidal signal are chosen as the reference force signal for experimental verification. Due to the mapping relationship between desired force and voltage, the valve’s control voltage is set to 1 V to generate the constant 10 kN reference force signal. When the disturbance is firstly set to a sinusoidal signal with 2 kN amplitude and 0.25 Hz frequency, the results of system force tracking performance with FASMC controller are shown in Fig. 20. Then the control voltage is changed to the sinusoidal form with 1 V amplitude and 0.5 Hz frequency under the same disturbance, thus, the results of system force tracking performance are shown in Fig. 21. From the comparative results in Figs. 20 and 21, the simulator system achieves a better tracking performance when inputting constant signal with the maximum tracking error 12.3 N, which is a little larger than the corresponding simulation result. When inputting sinusoidal signal, the maximum tracking error reaches approximately 63.7 N with slight jitter at the peak of the sinusoidal signal. Also, the convergence time under different reference signals are both within 0.8 s, which is faster than the simulation results.

Table 4 Specification of key components in simulator test rig. Component

Type

Specification

Hydraulic supply

YUKEN

Servo valve

MOOG

Hydraulic cylinder

Uranus

Force sensor

Futek

Displacement sensor

MTS-R

A/D card D/A card Computer

Advantech Advantech Advantech

System pressure: 21 MPa Max flow rate: 82 L/min Rated flow: 40 L/min Input voltage range: 5 to 5 V Stroke: 600 mm Max output force: 76.6 kN Range: 0  80 kN Accuracy: 4 N Range: 0  600 mm Accuracy: 5 lm PCI-1706 PCI-1711 IPC-610H

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Fig. 20. Test of force tracking performance under FASMC controller with a constant reference signal and small disturbance.

Fig. 21. Test of force tracking performance under FASMC controller with a sinusoidal reference signal and small disturbance.

To observe the robustness of proposed controller to high-frequency disturbance, the amplitude and frequency of disturbance are increased to 5 kN and 1 Hz, respectively. Therefore, different tracking performance under constant and sinusoidal control signal defined above are shown in Figs. 22 and 23, respectively. As shown in Figs. 22 and 23, with the severe disturbance (maximum amplitude 0.5 V and 1 Hz frequency), the maximum stable tracking errors rise to approximately 68.2 N and 117 N respectively under different reference signals, which are also larger than the simulation results because there are more uncertain nonlinearities not considered in the system model. In addition, the high-frequency disturbance leads to a degraded performance on the stability of system response, which can be reflected clearly by the severe fluctuation of output force signals. Despite larger tracking errors and fluctuation, the system still shows much better force tracking performance in real-time implementation compared with the conventional case A and control scheme I in case B according to Table 3. Moreover, the practical convergence time are turned out to be shorter than the simulation results, which also demonstrates the effectiveness of FASMC method with improved GMS friction compensation.

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137

Fig. 22. Test of force tracking performance under FASMC controller with a constant reference signal and large disturbance.

Fig. 23. Test of force tracking performance under FASMC controller with a sinusoidal reference signal and large disturbance.

6. Conclusion In this work, a finite-time adaptive sliding mode control strategy for electro-hydraulic load simulator with improved GMS friction characteristic is proposed for achieving the higher force tracking performance during the process of test for rocket thrust vector servo mechanism. An improved GMS friction model with parametric uncertainties is derived and identified by efficient PSO identification algorithm, which ensures a more accurate nonlinear friction description for control design. By employing an adaptive algorithm, uncertainties including part of friction, external disturbance and other unmodeled part are estimated effectively for compensation. Due to the high requirements for robustness and rapidity of load simulator, the proposed FASMC strategy combined with the improved GMS model based friction compensation can achieve a guaranteed rapidity and final tracking accuracy for real-time force loading under severe disturbance, which also indicates a wider range of applications involving electro-hydraulic system. Simulation and experimental results both prove a smaller tracking error and shorter convergence time of proposed FASMC tracking strategy compared to conventional compensation methods.

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