Experimental evaluation of the parameter-based closed-loop transfer

Special Issue Article

Experimental evaluation of the parameter-based closed-loop transfer function identification for electro-hydraulic servo systems

Advances in Mechanical Engineering 2017, Vol. 9(1) 1–12 Ó The Author(s) 2017 DOI: 10.1177/1687814016684425 journals.sagepub.com/home/ade

Guang-Da Liu1, Ge Li2,3 and Gang Shen2,3

Abstract Closed-loop systems of an electro-hydraulic servo system including position, acceleration, and force closed-loop systems and their closed-loop transfer functions based on parameter model are adaptive identified using a recursive extended least-squares algorithm. The position and force closed-loop tracking controllers are designed by a proportional–integral–derivative controller and are tuned by the position and force step signals. The acceleration closed-loop tracking controller is designed by a three-variable controller and the three states include position, velocity, and acceleration. Experimental results of the estimated position, acceleration, and force closed-loop transfer functions are performed on an actual electro-hydraulic servo system using xPC rapid prototyping technology, which clearly demonstrate the benefit of the adaptive identification method. Keywords Electro-hydraulic servo system, system identification, position closed-loop system, acceleration closed-loop system, force closed-loop system

Date received: 30 September 2016; accepted: 24 November 2016 Academic Editor: Nasim Ullah

Introduction A new structural diagram of electro-hydraulic servo system (EHSS) is shown in Figure 1, including position, acceleration, and force closed-loop servo systems, with advantages such as powerful density, large forces, and high tracking accuracy.1–3 The EHSS consists of a force loading system, a shaking table, a test specimen, eight exciters, two reaction walls and the portal frame, and so on. In order to achieve the vibration environment simulation, a test specimen is installed on the shaking table, which is connected to hydraulic exciters by hinge supports. The hydraulic exciters provide power to drive the shaking table. The EHSS has been employed in many position, vibration, and force tests to evaluate original structural performances and their potential problems under actual working environment.4–6 However, the acceleration/force loading tracking accuracy with

traditional methods on the EHSS is limited because of system nonlinearities including servo valve dynamics, spherical joint clearance, and frictions.7,8 In order to improve the tracking performance, the position, acceleration, and force closed-loop systems are designed to optimize the control methods of the EHSS. 1

College of Mechanical and Electronic Engineering, Eastern Liaoning University, Dandong, China 2 School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou, China 3 Jiangsu Key Laboratory of Mine Mechanical and Electrical Equipment, China University of Mining and Technology, Xuzhou, China Corresponding author: Gang Shen, School of Mechatronic Engineering, Jiangsu Key Laboratory of Mine Mechanical and Electrical Equipment, China University of Mining and Technology, Xuzhou 221116, China. Email: [email protected]

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).

2

Figure 1. Structural diagram of the EHSS.

A three-variable controller (TVC), including position, velocity, and acceleration, is commonly employed for acceleration tracking control,9 especially in E-Defense shaking table.10 In this article, the TVC is employed to control the acceleration closed-loop system, which can extend the frequency bandwidth and improve stability of the acceleration closed-loop control system. A proportional–integral–derivative (PID) controller is employed to implement force tracking control. After the position, acceleration, and force closed-loop systems have been established, their closed-loop transfer functions should be identified for further designing an advanced controller, such as feed-forward inverse model compensation based on the estimated transfer functions including the position,11 acceleration,6,9 and force closed-loop systems.6,12 In order to obtain an accurate feed-forward inverse model and improve tracking accuracy of some reference signals, it is important to ensure identification accuracy of the electro-hydraulic system transfer function. The currently widely used method is to identify the closed-loop system transfer function first, and then, their inverse transfer function is obtained by offline design method.6,9 System identification is an important approach to model dynamical systems, which includes non-parameter identification based on Welch’s periodogram method,13 a parameter identification algorithm based on recursive least-squares algorithm14,15 and a recursive extended least-squares (RELS) algorithm,6,9 a coefficient-level mapping method,16 a modalbased structural identification algorithm,17 and a statespace model algorithm,18 and so on. The identified model of non-parameter identification is the frequency characteristic, but the parameter identification can obtain model parameters, and it is difficult to design the inverse model using the non-parameter model. Liu and Lu19 developed a least-squares algorithm–based iterative

Advances in Mechanical Engineering identification method for a class of multi-rate sampleddata systems. Bao et al.20 proposed a least-squares algorithm–based iterative parameter estimation algorithm for multi-variable controlled autoregressive moving average (ARMA) model with finite measurement data. A standard least-squares identification algorithm was employed to identify the system parameters without considering the nonlinear dynamics characteristics.21 The RELS algorithm is an effective method to estimate the closed-loop system dynamics based on parameter model.22 Comparing with the least-squares algorithm, the RELS algorithm has a faster convergence performance and higher accuracy. In a study by Tang et al.,23 the RELS algorithm was applied to identify the electro-hydraulic system parameters, which was employed to design an inverse controller. In a study by He and Zhang,24 an RELS algorithm was proposed to identify parameters of the ARMA model of network utilization. In study by Rout and Subudhi,25 system parameters of an autonomous underwater vehicle were identified using the RELS algorithm. With comparison of the traditional fixed controller, the adaptive controller has strong adaptability characteristics, which can automatically adjust the controller parameters as the change of system performance. Chen et al.26 proposed a m-synthesis control strategy to improve the tracking performance of the linear motor driven stages with high-frequency dynamics for an adaptive robust control technique. In a study by Sun et al.,27 a constrained adaptive controller was designed to improve the vibration isolation performance for vehicle active suspension systems. Yao et al.28 proposed an adaptive backstepping controller with a friction compensator to improve the tracking accuracy with parametric uncertainties and nonlinear frictions. An adaptive repetitive controller was employed to compensate the periodic modeling uncertainties for motion control of the hydraulic servomechanisms.29 In order to improve closed-loop transfer function identification accuracy, the RELS algorithm is combined with adaptive control to identify the closed-loop transfer function of the EHSS, which is used to design the feed-forward inverse model. Experiments are performed on an actual experimental EHSS to verify the benefit of the identification method, and the experimental results demonstrate that the position, acceleration, and force closed-loop transfer functions based on parameter model can effectively be identified using the RELS algorithm. This article is organized as follows. Section ‘‘Dynamic model of the EHSS’’ describes the investigated EHSS and its nonlinear mathematical model. Section ‘‘Position, acceleration, and force controllers of the EHSS’’ discusses the position, acceleration, and force controllers of the EHSS. In section ‘‘Closed-loop transfer function identification algorithm,’’ the RELS algorithm is employed to identify the closed-loop system transfer functions. Experiments are

Liu et al.

3

conducted and their results are discussed to analyze the effectiveness of the identification algorithm in section ‘‘Experiment study.’’ Conclusions are drawn in section ‘‘Conclusion.’’

Dynamic model of the EHSS Figure 2. Block diagram of the displacement controller system.

Figure 1 shows the structure of the EHSS, which is comprised of eight hydraulic actuators controlled by eight servo valves. The model of the servo valve can be expressed as follows Gsv (s) =

xv Kv  =2 2jv s iA v2 + vv s + 1

where Ap is the effective area of the cylinder, y is the piston displacement, and y_ is the piston velocity. The total leakage coefficient Ctc can be written as Ctc = Cip + Cep =2, Cip and Cep are the internal and external leakage coefficients, respectively, KVB is the compress flow coefficient which can be defined by KVB = Vt =4be , Vt is the total volume of the cylinder, and be is the effective bulk modulus. The hydraulic actuator motion principle with force balance equation can be regarded as follows32

ð1Þ

v

where Q is the flow of the servo valve, iA is the servo valve drive current, xv is the displacement of the servo valve, Kv is the servo valve power amplifier gain, vv is the natural angular frequency of the servo valve, and jv is the damping ratio of the servo valve. The linearized load flow QL from the servo valve to the hydraulic actuator without considering the leak of the hydraulic actuator and compression of oil can be obtained as follows QL = Kq xv  Kc PL

where mt is the mass of the piston and the load, €y is the acceleration of the actuator piston, Bp is the viscous damping coefficient of the piston and the payload, and K is the force cell spring gradient. With combination of equations (2), (5), and (6), an open-loop dynamic model from the drive signal to the displacement of the hydraulic actuator without an external force with Laplace transformation can be expressed as follows

ð2Þ

where PL is the load pressure, defined by PL = P1  P2 , in which P1 and P2 are the pressure inside the two chambers of the cylinder. And the average load flow can be obtained as QL = Q1 + Q2 =2, in which Q1 and Q2 are the flow in and out of the servo valve, defining Kc is the flow pressure coefficient and Kq is a linearized flow gain coefficient, and can be expressed as follows30 ∂QL Kq = ∂xv Kc =

∂QL ∂PL

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ps  PL = Cd w r qffiffiffiffiffiffiffiffiffiffi L Cd wxv Ps P r = 2(Ps  PL )

G(s) = ð3Þ

y K0  = 2 xv h s vs 2 + 2z s + 1 vh

where K0 is the servo valve flow gain and is defined by K0 = Kq =A, vh is the natural angular frequency of the hydraulic actuator and can be represented by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vh = 4be A2 =Vt m, and zh is the damping ratio of the hydraulic actuator, which can be represented by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zh = Kce =A be m=Vt , and Kce = Kc + Ctc . And the block diagram of the displacement controller system is shown in Figure 2. According to equations (2), (5), and (6), the transfer function from the servo valve displacement to the actuator generated force can be written and simplified as follows

where Cd is the discharge coefficient, w is the constant area gradient of the servo valve orifices, r is the mass density of the fluid, and Ps is the hydraulic supply pressure, defined by Ps = P1 + P2 . With consideration of the total leakage coefficient, QL can be obtained by the flow continuity equation from the servo valve to the hydraulic actuator, given as follows31 ð5Þ

Kq Ap

(mt s2 + Bp s + K)   Vt Bp m t Vt 3 Kce mt 2 + 1 + Bp Kce + s + + s 2 2 2 4be Ap Ap 4be Ap A2p  2  K q Ap s 2jm Kce v2m + vm s + 1   = s s2 0 + 2j v0 s + 1 vr + 1 v2

GF (s) =

ð7Þ

h

ð4Þ

QL = Ap y_ + Ctc PL + KVB P_ L

ð6Þ

PL Ap = mt €y + Bp y_ + Ky



0

Vt K 4be A2p

 s+

Kce K A2p

ð8Þ

4

Advances in Mechanical Engineering

Table 1. Main parameters of the simulation model. Definition

Parameters

Values

Force cell spring gradient Linearized flow gain coefficient Flow pressure coefficient Effective area of the actuator Viscous damping coefficient of the piston and actuator Internal leakage coefficient External leakage coefficient Mass of piston and load Total volume of the actuator Effective bulk modulus Natural angular frequency of hydraulic actuator Damping coefficient of hydraulic actuator

K Kq Kc Ap Bc Cip Cep mt Vt be vh jh

23107 N=m 0:00145 (m3 =s)=V 231012 m3 =(s  Pa) 1:883103 m2 25, 000 N=(m=s) 4:631017 m3 =(s=Pa) 4:631017 m3 =(s=Pa) 5000 kg 0:963103 m3 6:93108 Pa 32 Hz 0.35

Figure 3. Block diagram of the force controller system.

where vm is the natural angular frequency of the paypffiffiffiffiffiffiffiffiffiffiffi load defined by vm = K=mt , jm is the damping ratio pffiffiffiffiffiffiffiffi of the payload defined by jm = Bp =(2 Kmt ), v0 is the natural angular frequency of the hydraulic actuator pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi defined by v0 = vh 1 + K=Kh , j0 is the damping ratio of the hydraulic actuator defined by j0 = 2be Kce Kh =½Vt (Kh + K)v0  + Bp =(2mt v0 ), vr is the first-order inertial element of the hydraulic actuating system defined by vr = Kce K=½A2p (1 + K=Kh ), and Kh is the liquid spring stiffness defined by Kh = mt vh . And the block diagram of the force controller system is shown in Figure 3. In order to evaluate the established model accuracy, the frequency response of the simulation and experimental results is presented in Figure 4, and the main simulation parameters are listed in Table 1. It can be seen that the mathematical model is closed to the actual experimental system.

Position, acceleration, and force controllers of the EHSS Position controller With consideration of the established dynamic model in equation (8), the position controller can be designed as Figure 5. The reference displacement signal rd (k) is sent to the electro-hydraulic position servo system, and the output feedback signal yd (k) is employed to adjust

Figure 4. Frequency characteristics of dynamic model and experimental results: (a) magnitude frequency characteristics and (b) phase frequency characteristics.

the drive signal. It is a simple position feedback system, and its closed-loop transfer function can be expressed as follows

Liu et al.

5

GAR (s) =

Kg + vg s + dg

G(s)Gsv (s) 1 + G(s)Gsv (s)

Kv Kd K0  2  =  2 s 2jsv s 2jh s 2 + s+1 + s + 1 + Kv Kd K0 vsv vsv vh v2h

W (s) =

ð9Þ where Kd is the displacement gain of the position closed-loop system.

G (s)   sv ðs=vd + 1Þ s2 v2nc + 2(jnc =vnc )s + 1

The TVC is an essential controller for electro-hydraulic shaking tables with 2-degree-of-freedom control structure including a feedback controller and a feed-forward controller. A block diagram of the TVC is given in Figure 6, where the design of the TVC is based on the position closed-loop system. The TVC directly represents the displacement and acceleration of the EHSS measured by a linear variable differential transformer (LVDT) and an accelerometer, respectively, while the velocity feedback signal is synthesized using a low-pass filter with the displacement and a high pass-filter with the acceleration. It can be seen that the acceleration reference signal ra (k) is converted into the displacement signal Rd by the acceleration generator and the TVC feed-forward controller. The acceleration and the velocity feedback signals are employed to compensate the position closed-loop controller, and the drive current signal of the servo valve from the position closed-loop controller is sent to the EHSS. In Figure 6, ya (k) and yd (k) are the actual output acceleration and the displacement signals, respectively. From Figure 6, it is obvious that the transfer function of the acceleration generator can be deduced as follows

 B(s) = Kdr

  2  Kdr Kar 2 s 2jnc 1+ s+ s = + s+1 Kdr Kdr v2nc vnc ð12Þ

where Kdr is the displacement feed-forward gain, Kvr is the velocity feed-forward gain, and Kar is the acceleration feed-forward gain. By combining equations (10), (11), and (12), the transfer function of the TVC can be described as follows Gac (s) = GAR (s)B(s)W (s) =

Kg Gsv (s) (s2 + vg s + dg )(s=vd + 1) ð13Þ

Force controller Figure 7 is a diagram of the force controller, where the PID controller Gcf (s) is employed for the closed-loop system to reduce the force tracking error of the EHSS. The reference force signal Fra is sent to the electro-

K ar

K vr

ra (k )

Kg

+ _ _

. . . 1 s

1 s

K dr

vg

+

+ +

EHSS Rd+

_

K df

+

+ _

_

K vf

Figure 6. Block diagram of the acceleration controller.

Gsv (s )

K af

dg

Acceleration Generator

ð11Þ

where vd is the desired acceleration frequency bandwidth, vnc is defined by vnc = 1:05vh ;1:2vh , and jnc ’ 0:7. The TVC feed-forward controller can be expressed as B(s) = Kdr + Kvr s + Kar s2 , and to cancel the poles of equation (7), the expression can be described as follows

Acceleration controller

TVC Feed-forward

ð10Þ

where vg = 2jn vn and dg = v2n are chosen to prevent the table from moving outside the maximum stroke of these cylinders when acceleration in a very low frequency bandwidth is applied, jn is the damping ratio of the acceleration control, vn is the original frequency of the acceleration control, and Kg is the acceleration control gain. Taking the dynamic characteristics of servo valves into consideration, the frequency response of the closed-loop system can be described as follows

Figure 5. Block diagram of the closed-loop position system.

Gpc (s) =

s2

TVC feedback

Velocity

G (s)

.. ya ( k )

y

yd ( k )

6

Advances in Mechanical Engineering

Figure 7. Block diagram of the force controller.

Closed-loop transfer function identification algorithm The closed-loop transfer function of the EHSS must be identified. The RELS algorithm is employed to identify the transfer function and improve closed-loop transfer function identification accuracy. The principle of frequency response function estimation based on the RELS algorithm is shown in Figure 8. ^ EHSS (z) is the identified transfer funcIn Figure 8, G tion, and the modeling error DGa is the difference between the identified model and the actual plant. It can be seen that the input signals, including the position reference signal rd (k), the acceleration reference signal ra (k), and the force reference signal rf (k), are sent to the RELS algorithm and the EHSS, respectively. The e(k) results from subtracting the outputs of the EHSS, including the position feedback signal yd (k), the acceleration feedback signal ya (k), the force feedback signal yf (k), and the ideal signals d(k) after reference model. In Figure 8(b), when the weight coefficient v(k) converges to an optimal solution, the error e(k) decreases, and y(k) is very close to the response of the RELS algorithm y0 (k). That is y(k) ’ y0 (k) Figure 8. Block diagram of the RELS algorithm: (a) RELS algorithm for the EHSS and (b) schematic representation of the RELS algorithm.

hydraulic force servo system, and the measured feedback force Fse is employed to adjust the input signal. As can be easily inferred from Figure 7, the transfer function of the force controller can be deduced as follows Gfc (s) =

Gcf (s)Gsv (s)GF (s) 1 + Gcf (s)Gsv (s)GF (s)

When the controlled object models are unknown, a general experience model is commonly used to describe the adaptive control strategy and it can be expressed as follows yk + 1 =

n X

ai, k g(yki ) +

i=0

m1 X

bj, k h(ukj )

ð16Þ

j=0

where y is the output signal, u is the input signal, g(y) and h(u) are the base functions, which can be linear

 s2 2jm + s+1 v2m vm   2    = 2 K q A p s2 s 2jv s s 2j0 2jm + s+1 +1 + s + 1 + Gcf (s)Kv + s+1 v2v vv v0 Kce v2m vm vr v20 K A Gcf (s)Kv Kqcep

ð15Þ



ð14Þ

Liu et al.

7

Figure 9. Experimental system.

function or nonlinear function, and a and b are unknown parameters to be identified. If the base functions g(y) and h(u) are linear functions, equation (16) can be translated to the ARMA model yk + 1 =

n X i=0

ai, k yki +

m1 X

bj, k ukj

ð17Þ

j=0

Because of the good convergence and simple calculation, the RELS algorithm is widely employed in realtime processing. However, there are noise and interference signals in the control system inevitably, and the identification of the noise model should be taken into consideration. Therefore, equation (17) is rewritten as follows yðk Þ = cT ðk Þu + zðk Þ

ð18Þ

where u = ½a1 , . . . , ana , b0 , . . . , bnb , c1 , . . . , cnc T , z(k) is a white noise sequence with zero mean, and c(k) is expressed by c(k) = ½y(k  1), . . . ,  y(k  na ), u(k), . . . , u(k  nb ), z(k  1), . . . , z(k  nc )T , ^j(k) is used to take the place of j(k) which cannot be measurable, and it can be expressed as follows ^ ^ ^ T (k)u(k) j(k) = y(k)  ^y(k) = y(k)  c

ð19Þ

T

^ a1 , ... ,^an , ^b0 , ... , ^bn ,^c1 , ... ,^cn  , c(k) ^ = where u=½^ a b c ½y(k  1), . . . , y(k  na ), . . . , u(k  d), . . . , u(k  d  nb ), ^ j(k  1) . . . , ^ j(k  nc )T . Similarly, the RELS algorithm can be expressed as follows6,9 h i ^ ðk Þ = u ^ ðk  1 Þ + K ðk Þ y ðk Þ  c ^ðk  1Þ ð20Þ ^ T ðk Þu u K ðk Þ =

^ ðk Þ Pðk  1Þc ^ T ðk ÞPðk  1Þc ^ ðk Þ r+c

ð21Þ

P ðk Þ =

i 1h ^ T ðk Þ Pðk  1Þ I  K ðk Þc r

ð22Þ

During the process of identification, the system ^ parameter u(k) and the covariance matrix need to be updated in every step of iterative operation, so two ^ parameters u(0) and P(0) should be given initial values before the beginning of an iterative operation. Furthermore, the covariance matrix must be the nonsingular matrix in order to ensure the convergence rate in iterative operation, expressed as P(0) = aI, where I represents identity matrix, and a should be a real large number to adjust the convergence rate of the system parameters.

Experiment study Experimental setup In this section, an experimental EHSS is shown in Figure 9, and Figure 10 shows how to implement the position, velocity, and acceleration controllers and their closed-loop transfer function identification. The experimental system consists of a platform, 8 actuating cylinders, a reaction base, 8 Moog two-stage servo valves (G761-3004), 16 spherical hinges, and some relative sensors including 8 LVDTs, 8 accelerometers, and 16 pressure transducers. Eight servo valves manufactured by Moog Company with a 38 L/min flow capacity at 7 MPa supply pressure control the cylinder. The actuators providing the driving forces have a 70mm bore and 50mm rod. The LVDTs are attached to the eight hydraulic actuators to measure the displacement and eight force sensors fixed between the hydraulic actuator for force measured and a spherical hinge to acquire the present force loading. Eight PCB accelerometers mounted on the drive rod of the eight cylinders are installed to obtain the acceleration information. The main parameters are listed in Table 2.

8

Advances in Mechanical Engineering

Table 2. Main parameters of the EHSS.

Table 3. Tuned experimental parameters.

Parameters

Values

Definition

Parameters

Values

Platform size Payload Control type Range of work frequency

2m32m 5000 kg Position, acceleration, and force Position: 0–20 Hz Acceleration: 1–50 Hz Force: 0–20 Hz 6 100 mm 0.2 m/s Non-loaded: 2.0 g Fully loaded: 1.0 g 1000 kg 6 1.5 t

Displacement gain Acceleration gain Acceleration feedback gain Velocity feedback gain Position feedback gain Acceleration feed-forward gain Velocity feed-forward gain Position feed-forward gain Original frequency of the acceleration control Damping ratio of the acceleration control

Kd Kg Kaf Kvf Kdf Kar Kvr Kdr vn

1.2 0.85 0.025 1 24.5 0.00053 0.84 1 3:14 rad=s

jn

0.7

Actuator stroke Maximum velocity Maximum acceleration Maximum load Range of force loading

EHSS: electro-hydraulic servo system.

A/D card PCI-1716 on the target computer. The control analog output signal completed by the D/A acquisition board ACL-6126 converts the digital signal into the analog signal by a signal modular and sends it to the servo valve for the control of hydraulic actuators.

Step experimental results The parameters of above-mentioned identification algorithm are adjusted with a step reference signal. As shown in Figure 12, these are experimental results of the position, acceleration, and force with their reference signals and experimental signals output, respectively, and the tuned experimental parameters are listed in Table 3. It can be obviously found that the performance of identification model using the step signal satisfies the actual experimental setup. So, the experiment of the tracking performance can be conducted. Figure 10. Control scheme of the EHSS.

The hardware architecture of the control system is processed on xPC target based on MATLAB/Simulink rapid prototyping. The programs of the proposed controller and data acquisition are designed with MATLAB/Simulink and compiled by Microsoft Visual Studio.NET on the host PC and then is downloaded to the target for implementation. As can be seen from Figure 11, the control hardware for the EHSS includes an Advantech IPC-610 controller, a PCI-1716, an ACL-6126, a host PC, and others. The acceleration and displacement signals are first converted and amplified by the conditioning module to 210 to + 10 V signals and then are collected by the 16-bit Advantech

Experimental results of the position closed-loop transfer function After the process of the position controller, the output signal collected could be used for system identification with the RELS algorithm. And, the reference signal is a random displacement signal with the magnitude of 0.5 g and the frequency range of 2–100 Hz; hence, the discrete transfer function of the position control system model can be expressed as follows Gp (z) = 0:02122z ^ 4  0:0922z ^ 3 + 0:1616z ^ 2  0:1324z + 0:04571 z ^ 4  3:457z ^ 3 + 4:825z ^ 2  3:165z + 0:8187

ð23Þ

Liu et al.

9

0.06 0.08 0.06 0.04 0.02 0 7.06

0.04 0.02 0 7.05

-4

x 10

Reference Experiemntal Output

0.08

7.065

7.07

7.075 x 10

7.1

4

7.15 7.2 Time(ms)

7.25

7.3 4 x 10

1 0 -1 -2 0

Reference Experimental output

0 -1

1.5105

1.511

1.5115 Time(ms)

1.512

1.5125

(b) 35

1.513 5 x 10

600

800

1000

7400

7600

7800 Time(ms)

8000

8200

8400

(c)

Figure 12. Experimental results with a step signal: (a) position control, (b) acceleration control, and (c) force control.

As shown in Figure 13, the estimated system model well matches the experimental result calculated by the RELS algorithm with a Hanning window and during the interest frequency for the magnitude and phase properties. Therefore, the conclusion can be drawn that the time and frequency response of the experiment is satisfied.

-10 -20

Identified model Experimental

-30 1

10 Frequency (Hz)

10

2

(b)

Reference Force Experimental output

30

0

-40 0 10

200

Phase(Deg)

Force(Kg)

400

(a)

1

25 7200

200

Time(ms)

Magnitude (dB)

Acceleration(g)

(a)

-2 1.51

Idetified model output Experimental output

2

Postion(m)

Displacement(m)

Figure 11. Experimental MATLAB/Simulink model for the EHSS.

100 0

Identified model Experimental

-100 -200 0 10

1

10 Frequency(Hz)

2

10

(c)

Figure 13. Experimental results of the position closed-loop control system: (a) time response, (b) magnitude frequency characteristics, and (c) phase frequency characteristics.

Experimental results of the acceleration closed-loop transfer function A random acceleration signal with the magnitude of 0.5 g and the frequency range of 2–100 Hz is employed in the acceleration control system, and the actual measured acceleration output signal is collected for system

Advances in Mechanical Engineering

26

Identified output Experimental output

0.2 0.1

25.5

Force(Kg)

Acceleration(g)

10

0 -0.1 -0.2 0

50

100 Time(ms)

150

24.5

200

24

5000

5200

10

5400 5600 Time(ms)

5800

6000

(a)

0

Identified model Experimental

0 Identified model Experimental

-20 -30 0

1

10

10 Frequency (Hz)

2

10

(b) 200

Magnitude (dB)

-10

100

-5 -10 -15

Identified model Experimental

0

20

40 Frequency (Hz)

60

80

60

80

(b)

0

200

-100 -200 0 10

1

10 Frequency(Hz)

10

2

(c)

Figure 14. Experimental results of the acceleration closedloop control system: (a) time response, (b) magnitude frequency characteristics, and (c) phase frequency characteristics.

Phase(Deg)

Magnitude (dB)

(a)

Phase(Deg)

Reference Identified model Experimental

25

Identified model Experimental

100 0 -100 -200 0

20

40 Frequency(Hz)

(c)

identification. The discrete transfer function of the TVC-controlled system model employing the abovementioned identification algorithm can be expressed as follows Ga (z) =

Figure 15. Experimental results of the force closed-loop control system: (a) time response, (b) magnitude frequency characteristics, and (c) phase frequency characteristics.

0:007585z ^ 6 + 0:05972z ^ 5  0:1888z ^ 4 + 0:3441z ^ 3  0:3756z ^ 2 + 0:2271z  0:05796 z ^ 6  5:342z ^ 5 + 12:25z ^ 4  15:43z ^ 3 + 11:24z ^ 2  4:484z + 0:7637

The time and frequency response of the experimental and identified EHSS is shown in Figure 14. From Figure 14, it can be obviously seen that the estimated system model well matches the experimental results calculated by the RELS algorithm with a Hanning window and during the interest frequency for the magnitude and phase properties. So, accuracy of the identified system model is reliable. Gf (z) =

24.25–25.75 kg and the frequency range of 0–20 Hz is applied to the PID controlled system, and the output signal from the experiment is collected for system identification. The discrete transfer function of the PID controlled system model can be expressed as follows

0:09572z ^ 6  0:5944z ^ 5 + 1:567z ^ 4  2:235z ^ 3 + 1:812z ^ 2  0:7909z + 0:1451 z ^ 6  5:382z ^ 5 + 12:17z ^ 4  14:79z ^ 3 + 10:19z ^ 2  3:769z + 0:5849

Experiment results of the force closed-loop transfer function After the EHSS is well-tuned by the PID controller, a sweep-frequency force signal with the magnitude of

ð24Þ

ð25Þ

To effectively estimate accuracy of the identified system model, the time and frequency response of the experimental and identified EHSS is depicted in Figure 15. The curve of the estimated system model well matches the experimental result calculated by the

Liu et al. RELS algorithm with a Hanning window and during the interest frequency for the magnitude and phase properties. Therefore, it can be concluded that the RELS algorithm is well to identify the force closedloop system.

Conclusion A dynamic model of the EHSS is established, and the position, acceleration, and force closed-loop controllers of the EHSS are designed, respectively. The acceleration closed-loop controller is designed by the TVC, and the position and force closed-loop controllers are designed by the PID controller. Closed-loop system transfer functions of the EHSS, including the position, acceleration, and force closed-loop system, are adaptive identified using the RELS algorithm. The experiments are conducted on hardware-in-the-loop-simulation platform using xPC rapid prototyping to verify the practical feasibility of the designed controllers and identification algorithm. In order to verify the benefit of the identification method, experiments are performed on an actual experimental EHSS, and the experimental results demonstrate that the position, acceleration, and force closed-loop transfer functions based on parameter model can effectively be identified using the RELS algorithm. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China (grant number 51575511) and Program for Changjiang Scholars and Innovative Research Team in University (No. IRT_16R68).

References 1. Yao JY, Jiao ZX, Ma DW, et al. High-accuracy tracking control of hydraulic rotary actuators with modeling uncertainties. IEEE/ASME T Mech 2014; 19: 633–641. 2. Wang CW, Jiao ZX, Wu S, et al. Nonlinear adaptive torque control of electro-hydraulic load system with external active motion disturbance. Mechatronics 2014; 24: 32–40. 3. Shen W, Mai YF, Su XY, et al. A new electric hydraulic actuator adopted the variable displacement pump. Asian J Control 2016; 18: 178–191. 4. Kenta S, Makoto I, Motohiro K, et al. Adaptive compensation for reaction force with frequency variation in shaking table systems. IEEE T Ind Electron 2009; 56: 3864–3871.

11 5. Lee SK, Park EC and Min KW. Real-time hybrid shaking table testing method for the performance evaluation of a tuned liquid damper controlling seismic response of building structures. J Sound Vib 2007; 302: 596–612. 6. Shen G, Zhu ZC, Li X, et al. Real-time electro-hydraulic hybrid system for structural testing subjected to vibration and force loading. Mechatronics 2016; 33: 49–70. 7. Yao JY, Jiao ZX and Ma DW. Extended-state-observerbased output feedback nonlinear robust control of hydraulic systems with backstepping. IEEE T Ind Electron 2014; 61: 6285–6293. 8. Yao JY, Jiao ZX and Ma DW. High dynamic adaptive robust control of load emulator with output feedback signal. J Frankl Inst 2014; 351: 4415–4433. 9. Shen G, Zhu ZC, Zhang L, et al. Adaptive feed-forward compensation for hybrid control with acceleration time waveform replication on electro-hydraulic shaking table. Control Eng Pract 2013; 21: 1128–1142. 10. Tagawa Y and Kajiwara K. Controller development for the E-Defense shaking table. Proc IMechE, Part I: J Systems and Control Engineering 2007; 221: 171–181. 11. Shen G, Lv GM, Ye ZM, et al. Implementation of electrohydraulic shaking table controllers with a combined adaptive inverse control and minimal control synthesis algorithm. IET Control Theory A 2011; 5: 1471–1483. 12. Zhao J, Shen G, Zhu W, et al. Robust force control with a feed-forward inverse model controller for electrohydraulic control loading systems of flight simulators. Mechatronics 2016; 38: 42–53. 13. Luh GC and Wu CY. Inversion control of non-linear systems with an inverse NARX model identified using genetic algorithms. Proc IMechE, Part I: J Systems and Control Engineering 2000; 214: 259–271. 14. Wang N, Sun J-C, Joo Er M, et al. Hybrid recursive least squares algorithm for online sequential identification using data chunks. Neurocomputing 2016; 174: 651–660. 15. Ziaei K and Sepehri N. Modeling and identification of electrohydraulic servos. Mechatronics 2000; 10: 761–772. 16. Jatinder S, Amitabh S and Vijay P. Application of system identification techniques to Tejas flight test data. Int J Aero Innovat 2013; 5: 11–18. 17. Suparno M, Hilmi L and Raimondo B. Modal parameter based structural identification using input–output data: minimal instrumentation and global identifiability issues. Mech Syst Signal Pr 2014; 45: 283–301. 18. Cara FJ, Juan J and Alarcon E. Estimating the modal parameters from multiple measurement setups using a joint state space model. Mech Syst Signal Pr 2014; 43: 171–191. 19. Liu XG and Lu J. Least squares based iterative identification for a class of multirate systems. Automatica 2010; 46: 549–554. 20. Bao B, Xu YQ, Sheng J, et al. Least squares based iterative parameter estimation algorithm for multivariable controlled ARMA system modelling with finite measurement data. Math Comput Model 2011; 53: 1664–1669. 21. Chen Z, Yao B and Wang Q. Accurate motion control of linear motors with adaptive robust compensation of nonlinear electromagnetic field effect. IEEE/ASME T Mech 2013; 18: 1122–1129.

12 22. Shen G, Zhu ZC, Li X, et al. Experimental evaluation of acceleration waveform replication on electrohydraulic shaking tables: a review. Int J Adv Robot Syst 2016; 13: 1–25. 23. Tang Y, Shen G, Zhu ZC, et al. Time waveform replication for electro-hydraulic shaking table incorporating off-line iterative learning control and modified internal model control. Proc IMechE, Part I: J Systems and Control Engineering 2014; 228: 722–733. 24. He F and Zhang H. Research on prediction of network utilization for NCS based on ARMA model and RELS. Intell Contr Autom 2014; 945–949: 2780–2783. 25. Rout R and Subudhi B. Inverse optimal self-tuning PID control design for an autonomous underwater vehicle. Int J Syst Sci 2017; 48: 367–375. 26. Chen Z, Yao B and Wang Q. m-synthesis based adaptive robust control of linear motor driven stages with highfrequency dynamics: a case study with comparative experiments. IEEE/ASME T Mech 2015; 20: 1482–1490. 27. Sun WC, Gao HJ and Kaynak O. Vibration isolation for active suspensions with performance constraints and

Advances in Mechanical Engineering

28.

29.

30.

31.

32.

actuator saturation. IEEE/ASME T Mech 2015; 20: 675–683. Yao JY, Deng WX and Jiao ZX. Adaptive control of hydraulic actuators with LuGre model-based friction compensation. IEEE T Ind Electron 2015; 62: 6469–6477. Yao JY, Jiao ZX and Ma DW. A practical nonlinear adaptive control of hydraulic servomechanisms with periodic-like disturbances. IEEE/ASME T Mech 2015; 20: 2752–2760. Lin TL, Huang WP, Ren HL, et al. New compound energy regeneration system and control strategy for hybrid hydraulic excavators. Automat Constr 2016; 68: 11–20. Liu Y, Gong GF, Yang HY, et al. Regulating characteristics of new tamping device exciter controlled by rotary valve. IEEE/ASME T Mech 2016; 21: 497–505. Guo Q, Zhang Y, Branko GC, et al. Backstepping control of electro-hydraulic system based on extended-stateobserver with plant dynamics largely unknown. IEEE T Ind Electron 2016; 63: 6909–6920.