PID TUNING FOR MULTI-CHANNEL STRUCTURAL FATIGUE TESTING SYSTEM

Hongwei , Shihui Duan and Jianmin Feng Aircraft Strength Research Institute of China [email protected]

Keywords: aircraft; multi-channel; control; PID;

Abstract Full-scale structural testing of aircraft structures has been and still is the task of the aircraft industry A proportion-integration-differentiation (PID) when new aircrafts are developed. In recent years, controller is a generic control loop feedback as China has been developing the aviation industry mechanism widely used in multi-channel aircraft and the research has been entering into a new stage, structural fatigue test. The PID parameter is the requirements of aircraft structure fatigue test obtained by test setup method in which PID is tuned become increasing. Aircraft structure fatigue test after the test setup is actually performed, so that the employ multi-channel coordinated loading control tuning period becomes overlong. And the dynamic system to drive the hydraulic system to apply load characteristic of mechanical and hydraulic spectrum to aircraft structure. Fatigue test system equipment isn’t considered as a crucial factor to adopts PID control. Namely, the testing system decrease working capability of loading equipment. calculates the control quantity by PID algorithm at the same time, there is coupling effect among based on system error. Currently, the tuning to PID channels, which lower control accuracy. For this, of test system is completed using experience after based on multi-channel fatigue test of cantilever test setup installation, so that the PID prediction is beam, the coupling mathematics model of testing difficult to implement to increase tuning time to system is presented and linearized reasonably. The impact test cycle. And dynamic characteristics of basic procedure of a multi-channel fatigue test is the mechanical and hydraulic equipment have not analyzed systematically. The PID parameter of considered fully, and the capacity of loading every channel is simulated by stability margin and equipment is not enough. Worse, the coupling bandwidth analysis. The dynamic characteristic of phenomenon among the actuator cylinders has been hydraulic equipment and test article are analyzed. occurring. Finally, a double channel fatigue test of cantilever Due to the above problems, aircraft structural test beam is implemented. The results of simulation and cannot be run smoothly, and the test results cycle test show that the new method is effective. This are affected seriously. For this, in 2010, Eenkhoorn method not only has the ability to predict PID N.C. of Delft University of Technology in the parameter rapidly and the dynamic characteristic of Netherlands has carried out the PID simulation in test equipment, but it can also reduce effectively the single channel structural fatigue[1]. Despite control errors. controller, hydraulic system and test article are 1 Introduce coupled, Dr E has not yet researched multi-channel test system, and the methods have not been applied 1 HONGWEI ZHAO, SHIHUI DUAN ＆ JIANMIN FENG

in engineering. In 2012, Xiasheng of aircraft fatigue test system as the research object. The strength research institute of China have studied the mathematical model of subsystem is presented. PID simulation of control channel for MTS control The coupling characteristic of test system is system[2]. This method is just focus on the analyzed. The PID parameter of every channel is controller, not the hydraulic system and test article. simulated to compensate coupling effect among In 2015, Shihui Duan of aircraft strength research channels. Dynamic characteristic of hydraulic institute of China has simulated PID parameter of system is described to improve capacity to testing the cantilever beam structure test system[3]. The system. control system, hydraulic system and mechanical system are coupled to tune PID of the testing 2 Modelling Theory system. This method is applied to structural tests. Unfortunately, the method is based on single 2.1 Modelling of servo-valve channel test system. The structure of the force feedback two stage servo The above researches are almost related to single valve is shown in Fig. 1. The valve keeps a good channel fatigue test system, so the multi-channel linear, due to Armature and Baffle working in the test system research is increasingly important. zero position. Therefore, in 2010, professor Huaihai of nanjing university of aeronautics and astronautics has studied variable parameter PID control method for multiple input multiple output random vibration test[4]. The basic procedure of multiple input multiple output random vibration test is analyzed systematically in the article. A definition of control error under logarithmic coordinates is provided, a new PID control strategy is introduced into the control algorithm. But the method does not take into account the hydraulic system and test article 1. Magnet 2. Lower guide magnet 3. Armature 4. Coil 5. Spring tube effecting to PID and control error. In 2013, 6. Upper guide magnet 7. Nozzle 8. Slide valve 9. Fixed orifice professor Jianyu Zhao of jinan university has Fig. 1 Principle diagram of servo vavle studied multi-channel coordinated loading control method[5]. This approach is based on neural There is a large number of parameters in the network algorithm. But it is not involving the structure of servo valve, so the dynamic hydraulic system and test article of testing system. characteristic of armature motor and nozzle flapper The other researching refer to the hydraulic is ignored. The paper just researches dynamic system[6~7] or control system[8~10] or developing characteristic of spool[20~21]. Its dynamic behavior to control algorithm[11~14] or the coupling is described by a second-order model： simulation between two systems[15~17] or the 1 2 xsv x  x  k I (1) 2 v v v sv sv coupling simulation among the hydraulic systems, sv sv control system and test article[18~19]. Currently, where ksv is valve gain; ωsv is eigenfrequency of the

The research about single-channel fatigue test is valve; ξsv is rate of damping coefficient; xv is more than multi-channel test system’s. So the thesis position of valve; Isv is valve control input. considers multi-channel cantilever beam structure Equations 1 is converted to transfer function by

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PID TUNING FOR MULTI-CHANNEL STRUCTURAL FATIGUE TESTING SYSTEM

Laplasse transform from valve control input to Combine equation 3,4,5 to obtain equation 6. spool position output, as Equations 2 shown:  1 qL1  x v P s  2    P L 2  (6) Xv() s k sv sv  Gs() (2)  1 22 1 2 Is  ss2sv  sv  sv    1   21 3   

2.2 Modelling of hydraulic actuator where ε1 is rate; ω is area gradient of valve orifice; P is supply pressure; ρ is oil density. The transfer function of hydraulic actuator is S The equation 6 can be linearized to gain equation 7 deduced using flow equation of valve, hydraulic by Taylor series. spring stiffness equation of actuator, flow continuity  equation of actuator and force balance equation of  qL k q11 x v k c P L piston[21~23]. Derivation process is divided into  (7)  1 two steps: spool moving to the left and right. kq11 C d  P s  2    P L    1 2  Cxdv  kc1   PP 2   s  

where kq1 is zero position flow gain when the spool

will move to the right; kc1 is flow-pressure

coefficient; Cd is flow coefficient;

When xv<0, the spool will move to the left, in the same way, equation 8 is obtained.  q k x k P  L q22 v c L  112 kq22 C d  P s  1   P L   (8) Fig. 2 structural diagram of Valve-controlled cylinder   1 2 2 1 Cxdv    k   As Fig. 2 shown, when spool position xv>0, the  c2  1 2 21PPsL   spool will move to the right. The flow rate of      2 actuator is equation 3.  1  2   21 3 QA22      (3)  QA11 AAA02 2 1   1 where α is flow rate(0.5≦α≦1); A1 is piston area where kq2 is zero position flow gain when the spool of rear chamber; A2 is anterior chamber. will move to the left; ε2 is rate; A01 is equivalent The load pressure can be written as equation 4. confined area of piston when the spool will move to  F PP the left. P 12  L (4)  A01 2   when hydraulic actuator system has its lowest AAA 1   01 1  2 eigenfrequency, its damping ratio is minimum, and where F is external load; A01 is equivalent confined the stability is worst, and the hydraulic spring area of piston when the spool will move to the right; stiffness is minimum.

PL is load pressure. The hydraulic spring stiffness is generated by the The load flow is defined as equation 5. compressibility of oil, when the chambers are seal QQ fully. Assuming that the piston rod length is x, q  12 (5) L 2 equation 9 is gained using Hooke's law. where qL is load flow.

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HONGWEI ZHAO, SHIHUI DUAN ＆ JIANMIN FENG

Fp11 equation 17. Kh1  (9) xxpp L VAA0 1 2  (17) where kh1 is hydraulic spring stiffness; F1 is 1 hydraulic spring force of rear chamber; Δp1 is At last, equation 18 is gained. pressure change of rear chamber; x is piston  p hV10 qL A e x p P L position.  2e  (18)  AA12 The bulk modulus βe of rear chamber is described as Ae   2  2   2   1    equation 10. h    1 2 2  1 10 1 p1  e  (10) VV11/ where Ae is average area of piston; h1 is Combine equation 9 and 10 to obtain equation 11. proportional coefficient.

2 In the same way, When oil flows out rear chamber, e A1 Kh1  (11) equation 19 is obtained. V1  hV20 qL A e x p P L In the same way, equation 12 is gained.  2e (19)  2  1  1 2 h    A  2 2 K  e 2 (12)  1 10 h2 V 2 For force balance equation of piston, When oil The hydraulic spring stiffness is described as equation flows into rear chamber, equation 20 is obtained. 13 for the whole hydraulic actuator system. F A01 PL mx p (20) 22 eeAA12 Kh  (13) VV12 When oil flows into rear chamber, equation 21 is obtained. When Kh is minimum, equation 14 is obtained.

F A02 PL mx p (21) AA22 12 (14) VV12 Not considering external load, transfer function of actuator is gained, as equation 22 shown. Finally, when hydraulic actuator system has its F(s )ksv 1 (22) lowest eigenfrequency, piston position x is Gs2 () p Xs() hV m h ms2 v 10 s32 mk1 s A s described by equation 15. ce 2e A01 h 2 A 01 A 02 L x  (15) where F(s) is output loading; m is mass of dynamic p 1 section. Where L is stroke of actuator. Piston operation is divided into two steps: flowing 2.3 Modelling of test article into rear chamber, flowing out anterior chamber; flowing out rear chamber, flowing into anterior To couple test article with hydraulic system, the test chamber. article state space model is presented. Under When oil flows into rear chamber, equation 16 is physical coordinates, the dynamic model of test obtained. article is written as equation 23.

QQAAVV1 2 1 2 1 2 (16) qL   y  P12  P Mww t  C t  Kw()() t  Fe t (23) 2 2 2ee 2

And total volume of actuator Vo is described as where w(t) is displacement vector of test article; Fe(t) is external force; M is mass matrix; K is 4

PID TUNING FOR MULTI-CHANNEL STRUCTURAL FATIGUE TESTING SYSTEM

Stiffness matrix; C is damping matrix; 2 1 293 kp s293 k i  k p s  k i By introducing state space model, the number of Is  2.3 2.3 (26) Gs4 () states is increased by equation 23, which increases Es  2 1 s293 kd s the computational time. To decrease the number of 2.3 states it is possible to perform a modal analysis and where I(s) is control input; E(s) is control error; P is obtain a state space representation using proportional gain; I is integral gain; D is damping eigenfrequencies and eigenvectors, which is called a gain. normalized state space[24]. To simplify the complexity of the model and reduce computational time, reduction of the normalized state space model can be applied. equation 24 is obtained.

 xx 0 I   0  zu     22      (24)  x m 2  m m  x   r() t    m 0 x 0 Fig. 3 Block diagram of MOOG controller        0 m x 0 Finally, based on transfer function of each where x is generalized displacement; Ωm is subsystem, transfer function, Gk(s) for open loop eigenfrequencies matrix; ζm is structural damping system and Gb(s) for closed loop system, can be matrix; Φm is mode shape vector; r(t) is input load gained, as equation 27 shown. matrix.

In fact, Ωm, ζm and Φm are related to M, K and C  Ws  Gk()()()()() s G 4 s G 1 s G 2 s G 3 s which are obtained by Finite element model of test  Es   (27) article. Gs() Gs() k  b The state space equation 24 can be converted into  1 Gsk ( ) transfer function 25. Equation 27 can be analyzed and simulated using Ws  1 (25) G3(s)   C 2m2m sI 2m2  m  A 2m2  m  B 2m1   D 2m1  stability margin analysis and bandwidth analysis. Fs  Namely, keep gain margin h>0, phase margin θ>0 where W(s) is displacement of loading point; F(s) is and bandwidth B>10 Hz to simulate testing system. input load. The gain and phase margin are measurement to The key to study PID tuning method of relative stability, and bandwidth is measurement to multi-channel testing system is presenting to track performance of reference signal[25]. coupled mathematical model of test article. The coupling among channels is related to coupling 3 Simulation and Analysis among loading points directly. A double-channel fatigue testing system of 2.4 Modelling of controller cantilever beam is used to verify the new mothod.

In simulation, assume that kp=10, ki=0.18, kd=0.1, MOOG control system which widely used in dynamic characteristic of controller is shown as Fig. structural fatigue test is selected as controller in the 4. paper. Block diagram of MOOG controller is shown as Fig. 3. Equation 26 is obtained by the block diagram.

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HONGWEI ZHAO, SHIHUI DUAN ＆ JIANMIN FENG

Bode Diagram Bode Diagram 70 160 P Control 60 140 PI Control 50 PD Control 120 40

30 100

20 80 Magnitude (dB) Magnitude 10 (dB) Magnitude 60 0 40 -10

-20 20 90 -90

45 -135

0 -180

Phase (deg) Phase Phase (deg) Phase -45 -225

-90 -270 -3 -2 -1 0 1 2 3 -1 0 1 2 3 10 10 10 10 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Fig. 4 Linearized frequency response of Controller Fig. 6 Bode plot, presenting linearized frequency dependent behaviour of actuator I for a valve position input and piston MOOG 3002B valve is selected as the simulated position output servo valve in the paper. Its parameters are as follows. K =0.0625; ω =75 Hz; ξ =0.7063. sv sv sv Bode Diagram 160 Based on parameters and equation 2, transfer 140 120

100

function and bode plot of 3002B valve are obtained, 80 Magnitude (dB) Magnitude 60 as equation 28 and Fig. 5 shown. 40 20 13879 -90 Gs() (28) -135 1 2

ss666 222070 -180 Phase (deg) Phase -225

Bode Diagram -20 -270 -1 0 1 2 3 10 10 10 10 10 -40 Frequency (Hz) -60 -80 Fig. 7 Bode plot, presenting linearized frequency dependent

Magnitude (dB) Magnitude -100

-120 Motion dynamics input current output valve postion behaviour of actuator J for a valve position input and piston -140 0 position output -45

-90 Phase (deg) Phase -135  6729 1 Gs2，I () -180  3 2 2 -1 0 1 2 3 4 5 10 10 10 10 10 10 10  1.2e 9ss  7.4e  9  0.0022s 13.65s Frequency (Hz)  (29)  6647 1 Fig. 5 Servo valve motion dynamics, between current Gs2，J ()  1.7e 9ss3  6.9e  9 2  0.003s 18.5s 2 input and position output A two meters cantilever beam is selected as the Hydraulic actuator contains two model. Actuator I is test article in the simulation. Its scale is 3 -6 2 tonnage and 300 mm stroke, actuator J is 3 0.1×0.1×0.005 m; ρT=7580 Kg/m ; IT=2.8568×10 tonnage and 300 mm stroke. Parameters of actuator m4; E=210 GPa; Midpoint and endpoint of test 2 2 I are as follows. A1=0.0012 m ; A2=0.0031 m ; article is loading. 2 m=13.65 Kg; for actuator J, A1=0.0016 m ; The current state space model is calculated from a 2 A2=0.0044 m ; m=18.5 Kg; βe=1.3 GPa; ρoil=900 finite element model using 100 elements in total, 3 Kg/m ; Ps=21 MPa; Cd=0.7; because the interest is on the low frequency Based on parameters and equation 22, transfer depended behaviour up to 1000 Hz. As a result 200 function and bode plot of actuator I and J can be eigenfrequencies are obtained from the eigenvalue obtained, as equation 29 and Fig. 6,7 shown. analysis, since M-1K results in a 200×200 matrix. 200 Eigenfrequencies and a 200×200 eigenvector matrix is large (for our application), since fatigue testing is performed in a frequency range till 5 [Hz]. Reduction of the state space system is needed to

6

PID TUNING FOR MULTI-CHANNEL STRUCTURAL FATIGUE TESTING SYSTEM

reduce computational time. For the reduction step, sufficient because the fatigue loading reference model truncation is used, taking into account the signal has a low frequency content. Transfer first 4 eigenmodes and eigenfrequencies of the function and bode plot of test article are obtained, as system. This resulted in an accurate description of equation 30 and Fig. 8 and 9 shown. the system behaviour up to 100 [Hz], which is

 1.14e-13s7 +0.49s 6 +84s 5 +2e7s 4 +1.41e9s 3 +1.2e14s 2 +1.41e15s+4.11e19 G3,I (s)   209s7 +4.77e7s 6 +4.88e9s 5 +4.12e14s 4 +1.37e16s 3 +4.46e20s 2 +1.91e21s+1.35e25  7 6 5 4 3 2  -8.5e-14s -0.1s -22s -5.8e6s -4.4e8s -4.5e13s -1.42e14s-1.12e19 (30) G3,J (s)   209s7 +4.77e7s 6 +4.88e9s 5 +4.12e14s 4 +1.37e16s 3 +4.46e20s 2 +1.91e21s+1.35e25

Bode Diagram Bode Diagram -60 -80

-80 -100

-100 -120 -120 -140

-140 Magnitude (dB) Magnitude Magnitude (dB) Magnitude -160 -160

-180 -180 Reduced Modal State Space 1% damping Reduced Modal State Space 1% damping -200 -200 0 360

-45 180

-90 Phase (deg) Phase Phase (deg) Phase 0 -135

-180 -180 -1 0 1 2 3 -1 0 1 2 3 10 10 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Fig. 8 Cantilever Beam bode plot for point I Fig. 9 Cantilever Beam bode plot for point I

Two load applied on the test article are both down, valve, hydraulic actuator, and test article are which formed a strong coupling state between two coupled to obtain transfer function and bode plot of loading points. So transfer function of point I and J controlled system, as equation 31 and Fig. 10 and is relevant. PID tuning method can be used to 11 shown. decrease the relevance. For channel I and J, servo

 0.32e6s7 +7.3e11s 6 7.5e12s 5 +6.3e17s 4 2.1e19s 3 6.86e23s 2 2.94e24s+2e28 G123,I (s)   1.9e-22s12 +8.16e-10s 11 +6.87e-7s 10987 +0.04s +26s +2.7e5s +1.8e8s 6 +4.9e11s 5 +2.9e14s 4 +2.21e17s 3 +8.31e19s 2 +2.74e22s  7 6 5 4 3 2 (31)  0.32e6s +7.3e11s 7.5e12s +6.3e17s 2.1e19s 6.86e23s 2.94e24s+2e28 G123,J (s)   -1.42e-22s12 -1.67e-10s 11 -1.48e-7s 1098 -0.01s -7.48s -0.95e5s 7 -0.64e8s 6 -1.38e11s 5 -7.73e13s 4 +8.1e15s 3 +2.25e19s 2 +7.46e21s

Bode Diagram Bode Diagram 140 140

120 120

100 100

80 80 Magnitude (dB) Magnitude Magnitude (dB) Magnitude 60 60

40 40 Linearized FR between valve input and force output Linearized FR between valve input and force output 20 20 90 135

0 90

-90 45

-180 0 Phase (deg) Phase Phase (deg) Phase -270 -45

-360 -90

-450 -135 -1 0 1 2 3 -1 0 1 2 3 10 10 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Fig. 10 Frequency response of Controlled system(I) Fig. 11 Frequency response of Controlled system(J)

Based on stability margin analysis and bandwidth P=1.4~2, I=2.8~4, D=0.02~0.05. A group of theory, PID parameter of fatigue testing system is parameter in the each scope is selected to obtain tuned to reduce testing time and cost and risk. transfer function and bode plot of open and closed For channel I, PID is tuned as follows. P=1.5~2.5, loop system, when P=2, I=4, D=0.02 for the I=3.5~5, D=0.015~0.03. And for channel J, channel I, P=1.8, I=3, D=0.04 for the channel J, as

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HONGWEI ZHAO, SHIHUI DUAN ＆ JIANMIN FENG

equation 32 and Fig. 12 and 13 shown.

 2.6s9 +6e5s 8 +4e8s 7 +5e12s 6 +3e15s 5 +5.6e18s 4 +3e21s 3+2e23s 2 +10e25s+4e25 G (s)=  k,I 14 13 12 11109 8 7 6 5 4 3 2  1.43e-22s +6.16e-10s +5.2e-7s +0.03s +20s +2e5s +1.4e8s +3.6e11s +2e14s +1.6e17s +6e19s +2e22s +1.23e23s  2.6s9 +6e5s8 +4e8s 7 +5e12s 6 +3e15s 5 +5.6e18s 4 +3e21s 3 +2e23s 2 +10e25s+4e25 Gb,I (s)= (32)  1.43e-22s14 +6.16e-10s 13 +5.2e-7s 12 +0.03s 11109 +20s +2e5s +1.4e8s 8 +3.6e11s 7 +2e14s 6 +1.6e17s 5 +6.6e19s 4 +2e22s 3 +3e23s 2  9.8e25s 4.25e25  9 8 7 6 5 4 3 2  2.3s +5e5s +3e8s +4.6e12s +2.38e15s +5e18s +2.43e21s +1.6e23s +7.2e25s+3.17e25 Gk,J (s)=  -1.43e-22s14 -1.67e-10s 13 -1.5e-7s 121110 -0.01s -7.6s -0.95e5s 9 -0.64e8s 8 -1.4e11s 7 -7.9e13s 6 +7.18e15s 5+2.26e19s 4 +7.75e21s 3 +9.08e23s 2   2.3s9 +5e5s 8 +3e8s 7 +4.6e12s 6 +2.38e15s 5 +5e18s 4 +2.43e21s 3 +1.6e23s 2 +7.2e25s+3.17e25 Gb,J (s)=  -1.43e-22s14 -1.67e-10s 13 -1.5e-7s 121110 -0.01s -7.6s -0.95e5s 9 -0.64e8s 8 -1.4e11s 7 -7.44e13s 6 +9.55e15s 5 +2.76e19s 4 +1.02e22s 3 +2.5e23s2  7.29e25s 3.17e25

As Fig. 12 shown, h=1.12 dB, θ=0.539 deg. Channel I is stable, While B=15 Hz, tracking performance of control signal is well. In the same way, As Fig. 13 shown, h=8.75 dB, θ=3.71 deg. Channel J is stable too, While B=18 Hz, tracking performance of control signal is better than channel I. In fact, due to fault tolerance to controller parameter, it is reasonable that the predicted PID is in the scope.

Bode Diagram Gm = 1.12 dB (at 11.1 Hz) , Pm = 0.539 deg (at 10.5 Hz) 80

60

40

20

0

-20 Magnitude (dB) Magnitude -40 Closed Loop system -60 Open Loop system -80 0

-180 -360 -540

Phase (deg) Phase -720 -900 Fig. 14 Platform of cantilever beam fatigue test -1080 -1 0 1 2 3 10 10 10 10 10 Frequency (Hz) The reference signal is sinusoidal signal in the Fig. 12 Dynamic curve of channel I channel I and J. The signal amplitude is 3 KN and

Bode Diagram Gm = 8.75 dB (at 18.9 Hz) , Pm = 3.71 deg (at 13.6 Hz) 4.5 KN respectively, phase is zero, and frequency is 80

60 40 0.1 Hz. 20 0 Test result is as follows. Control error is given in

-20 Magnitude (dB) Magnitude -40 Closed Loop system -60 method E, Sun, Zhao, and test setup method and the Open Loop system -80 135 90 new method, as Fig. 15 to 24 shown. Tab. 1 shows 45

0 -45 contrasting to load error in several methods.

-90 Phase (deg) Phase -135 -180 0.8

-225 -1 0 1 2 3 10 10 10 10 10 0.6 Frequency (Hz) 0.4

Fig. 13 Dynamic curve of channel J 0.2

0 (%) 4 Test and Verification -0.2 -0.4

-0.6 A platform of double-channels cantilever beam -0.8 Channel I load error -1 structural fatigue test as Fig. 14 shown, is used to 0 5 10 15 20 25 30 time(sec) verify the PID tuning method. Fig. 15 Error of method E in the channel I

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PID TUNING FOR MULTI-CHANNEL STRUCTURAL FATIGUE TESTING SYSTEM

0.8 1

0.6 0.8

0.6 0.4

0.4 0.2

0.2 0

0

(%) (%) -0.2 -0.2

-0.4 -0.4

-0.6 -0.6

-0.8 -0.8 Channel I load error Channel J load error -1 -1 0 5 10 15 20 25 30 0 5 10 15 20 25 30 time(sec) time(sec) Fig. 16 Error of method Sun in the channel I Fig. 20 Error of method E in the channel J

0.6 1

0.4

0.5 0.2

0

0

(%) (%)

-0.2

-0.5 -0.4

-0.6 -1 Channel I load error Channel J load error

0 5 10 15 20 25 30 0 5 10 15 20 25 30 time(sec) time(sec) Fig. 17 Error of method Zhao in the channel I Fig. 21 Error of method Sun in the channel J

0.8 0.3

0.6 0.2

0.4

0.1 0.2

0

0

(%) (%)

-0.1 -0.2

-0.4 -0.2

-0.6 -0.3 Channel I load error Channel J load error -0.8 0 5 10 15 20 25 30 0 5 10 15 20 25 30 time(sec) time(sec) Fig. 18 Error of test setup method in the channel I Fig. 22 Error of method Zhao in the channel J

0.3 0.4

0.2 0.3

0.2

0.1 0.1

0 0

(%) (%)

-0.1 -0.1

-0.2

-0.2 -0.3

Channel I load error -0.4 Channel J load error -0.3 0 5 10 15 20 25 30 0 5 10 15 20 25 30 time(sec) time(sec) Fig. 19 Error of the new method in the channel I Fig. 23 Error of test setup method in the channel J

9

HONGWEI ZHAO, SHIHUI DUAN ＆ JIANMIN FENG

0.4 Methodology for Structural Fatigue Testing Setups[D].

0.3 The Netherlands. Delft University of Technology, 0.2 2010. 0.1 [2] . Method for PID Multi-objective

0 (%)

-0.1 Optimization for Structural Fatigue Test[D]. Bei Jing:

-0.2 Chinese Aeronautical Establishment. 2012.

-0.3

Channel J load error [3] Zhao Hongwei. Study of Simulated Technology for -0.4 0 5 10 15 20 25 30 time(sec) Structural Fatigue testing system[D]. Bei Jing: Chinese Fig. 24 Error of the new method in the channel J Aeronautical Establishment. 2015. [4] Xuli, Chen Huaihai, He Xudong et al.. PID control Tab. 1 contrasting to load error with variable arguments for MIMO random vibration

Method Method Method Test setup The new test[J]. Acta Aeronautica et Astronautica Sinica, Error E Sun Zhao method method 2010,31(9): 1776-1780.

Error I 0.85% 0.9% 0.7% 0.35% 0.29% [5] Yang, Zhao Jianyu, Xudong.

Error I 0.93% 1.2% 0.78% 0.42% 0.37% Multi-channel synchro-loading control meyhod[J]. Journal of university of JINAN (sci & Tech), 5 Final considerations 2013,27(2): 136-139. [6] Dong Haoran. Simulation research on hydraulic In first two methods, in the one hand, coupling system of marine steering gear[D]. Jian. Jimei between channels is not considered, so the control University.2009. error is larger than others obviously. In the another [7] Xiaolin, Kun. Design and Simulation of hand, due to hydraulic system and test article controller for large load hydraulic loading system[J]. coupled, the error of method E is lower than method Control Engineering of China, 2014,21(2):210-214. Sun’s. The error of the new method is the lowest [8] Shi Zhenyun, Wang Tianmiao, Liu Da et al. A fuzzy than others. In general, PID of testing system is PID-controlled SMA actuator for a two-DOF joint[J]. predicted using computational models before the Chinese Journal of Aeronautics. 2014,27(2):453–460. certification test is actually performed in the new [9] Liu Huibo,Wang Jing, Yanhe. Research and method. Simulation of fuzzy adaptive PID control for brushless  The testing behaviour and PID parameter of DC motor[J]. Control Engineering of China, testing system is predicted using computational 2014,21(4):583-587. models before the certification test is actually [10] Yang Zhi, Chen Zhitang, Fan Zhengping et al. Tuning performed in the new method; of PID controller based on improved  The control error of the new method is partical-swarm-optimization[J]. Control Theory & minimum in all of the methods to decrease Applications, 2010, 27(10): 1345-1352. couple affect among channels. [11] Jinda,Qi Jianhong. ARM9 based adaptive fuzzy  The new method provide not only theoretical PID constant cutting control system[J]. Control basis to PID tuning of multi-channel fatigue Engineering of China, 2014,21(3):315-320. testing system, but a methodology to similar [12] Zhiqiang, Jiang Ziya, He Yuqing et al. Frequency systems. properties of PID controller and model free tuning[J]. Control and Decision,2014,29(10): 1833-1838. References [13] Haibo, WANG Jiankang, CHEN Guoqiang et

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