Hybrid Robust Control Law with Disturbance

applied sciences Article

Hybrid Robust Control Law with Disturbance Observer for High-Frequency Response Electro-Hydraulic Servo Loading System Zhiqing Sheng † and Yunhua Li *,† School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China; [email protected] * Correspondence: [email protected]; Tel./Fax: +86-10-8233-9038 † These authors contributed equally to this work. Academic Editor: Chien-Hung Liu Received: 23 December 2015; Accepted: 28 March 2016; Published: 6 April 2016

Abstract: Addressing the simulating issue of the helicopter-manipulating booster aerodynamic load with high-frequency dynamic load superimposed on a large static load, this paper studies the design of the robust controller for the electro-hydraulic loading system to realize the simulation of this kind of load. Firstly, the equivalent linear model of the electro-hydraulic loading system under assumed parameter uncertainty is established. Then, a hybrid control scheme is proposed for the loading system. This control scheme consists of a constant velocity feed-forward compensator, a robust inner loop compensator based on disturbance observer and a robust outer loop feedback controller. The constant velocity compensator eliminates most of the extraneous force at first, and then the double-loop cascade composition control strategy is employed to design the compensated system. The disturbance observer–based inner loop compensator further restrains the disturbances including the remaining extraneous force, and makes the actual plant tracking a nominal model approximately in a certain frequency range. The robust outer loop controller achieves the desired force-tracking performance, and guarantees system robustness in the high frequency region. The optimized low-pass filter Q(s) is designed by using the H8 mixed sensitivity optimization method. The simulation results show that the proposed hybrid control scheme and controller can effectively suppress the extraneous force and improve the robustness of the electro-hydraulic loading system. Keywords: electro-hydraulic servo loading system; extraneous force; feed-forward compensation; disturbance observer; robust control

1. Introduction The electro-hydraulic servo loading system is a hydraulic force (moment) control system which is used to apply the requested load to the actuating component or the manipulation component. For example, the experiment on the hydraulic booster in the helicopter-manipulating system needs this system. The hydraulic booster is in charge of transmitting the manipulation displacement signal and amplifying the manipulating force; meanwhile, it must respond to the driver’s command with high accuracy and fast speed under the high-frequency aerodynamic load. Through the electro-hydraulic loading system, which is employed to simulate the aerodynamic load on the ground, the control performance with the load of the booster can be evaluated, and this has vital significance in product performance experiments and improvement. There are two types of loading: active and passive. The difficulty in the passive loading system is achieving the desired loading force on the condition of accompanying the motion of the loaded plant. There exists an unavoidable problem in such a loading system, namely how to restrain the motion

Appl. Sci. 2016, 6, 98; doi:10.3390/app6040098

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disturbance (the so-called extraneous force [1]), which originates from the motion of the loaded plant. This disturbance seriously affects the force-tracking accuracy, reduces the closed-loop bandwidth of the loading system, and seriously causes gradation in the force control performance. Therefore, extraneous force suppression is the key issue to ensure the reliability and confidence level of the experiment. As noted by Alleyne and Liu [2], for hydraulic loading systems, the tracking performance is limited by flow nonlinearity, parametric uncertainty, and motion disturbance, and cannot be guaranteed with a simple PID control algorithm. Thus, the online self-tuning fuzzy PID [3] and the grey predictor fuzzy PID [4] control algorithm were implemented, respectively. Robust control technologies, such as quantitative feedback theory (QFT) [5,6], the H8 mixed sensitivity method [7], variable structure control [8,9], neural network control [10,11], and fractional order adaptive robust control [12] had been investigated for robust force tracking in the presence of disturbances and uncertainty. Focusing on nonlinear characteristics and parameter uncertainty, some nonlinear system control methods [13–16] had also been investigated. In the above studies, the extraneous force had been treated as a part of the loading error, and the force-tracking performance was obtained through feedback and feed-forward controllers. From the forming mechanism of the extraneous force and the viewpoint of disturbance compensation, Liu [1] in China proposed a disturbance compensation method based on the structure invariance principle, actuator velocity was used to eliminate the actuator’s motion disturbance. This feed-forward compensation belongs to the direct disturbance compensation. Although it had made good elimination results in a low-frequency loading system, there were problems in physical implementation and velocity acquisition. Moreover, there is a lack of robustness against system uncertainty. In order to overcome the difficulty in acquiring velocity, a velocity synchronization method [17] was proposed, where the position input of the servo valve was alternatively utilized as a velocity signal to design the compensator. In view of improving the robustness of the compensation term, a hybrid control method [18] and dual-loop control [19] had further been proposed. Considering system uncertainties, the disturbance observer–based controller with a double-loop structure [20,21] was investigated. Problems of disturbance suppression and force tracking had been solved separately: the inner loop compensator was designed to restrain the disturbances and perturbations and to make the actual plant approximate the given nominal model, and the overall performance of the system was improved by the outer loop controller. So far, some compound control schemes combined with feed-forward compensation, such as double-loop cascade composition control [22], the nonlinear QFT technique [23], inverse model observer [24], adaptive control law [25], the H8 method [26], and iterative learning [27], etc., had been put forward. The known studies show that force-tracking accuracy is limited by the extraneous force suppression level, especially in the presence of parameter perturbation. With the continuously increasing performance requirements for the loading system, it is difficult to obtain better tracking performance with a single approach, even if such an approach had made achievements in the low-frequency case. The electro-hydraulic system is of high nonlinearity and parameter uncertainty. Nonlinearity mainly is caused by the flowrate-pressure characteristics of the servo valve, the compressibility of the hydraulic oil and the friction force of the hydraulic cylinder. A four-quadrant flowrate-pressure nonlinear model of the servo valve was established in [28–31]. Kemmetmüller and Kugi studied the impedance-based nonlinear control of the hydraulic servo system [28]. Scheidl and Manhartsgruber studied the singular issue in the mathematic model of the nonlinear hydraulic drive system, and proposed an approach of two subsystems with a boundary layer to analyze the nonlinear behavior [29]. Sun and Chiu studied the design of the nonlinear disturbance observer–based hydraulic force controller, and proposed the observer with a simple PI structure and a nonlinear control law based on the passivity theorem [30,31]. Currently, problems exist in the application of the above-mentioned nonlinear control approach in the force servo system, and the exact mathematical model of the servo valve is needed. Moreover, the robust control law, which is based on the Q-filter disturbance observer and the H8

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theory and does not rely on the exact mathematical model, can be easily designed. Therefore, this paper mainly studies the design method of this kind of control law. Appl. Sci. 2016, 6, 98  3 of 27  This paper focuses on the high-frequency loading issue of an electro-hydraulic loading systemThis paper focuses on the high‐frequency loading issue of an electro‐hydraulic loading system  with high-frequency dynamic load superimposed on a large static load of the helicopterwith  high‐frequency  dynamic  superimposed  on  a control large  scheme static  load  the  helicopter‐  manipulating booster, and mainlyload  studies a linear hybrid withof a constant velocity manipulating booster, and mainly  studies a linear  hybrid control scheme with a constant velocity  feed-forward compensator and a double-loop cascade composition control strategy. The constant feed‐forward  compensator  and  a  double‐loop  control force. strategy.  The  constant  velocity compensator is designed to eliminatecascade  most composition  of the extraneous The disturbance velocity  compensator  designed  to restrains eliminate  of  the  extraneous  The  disturbance  observer–based inner loopis compensator themost  disturbances, including force.  the remaining extraneous observer–based  inner  loop  compensator  restrains  the  disturbances,  including  remaining  force, and makes the actual plant tracking a nominal model approximately in a certainthe  frequency range. extraneous force, and makes the actual plant tracking a nominal model approximately in a certain  The robust outer loop controller guarantees the desired force-tracking accuracy, and improves system frequency  range.  The  robust  outer  loop  controller  guarantees  the  desired  force‐tracking  accuracy,  robustness in the high-frequency region. and improves system robustness in the high‐frequency region.  2. System Overview and Analysis 2. System Overview and Analysis  2.1. System Description 2.1. System Description  The helicopter-manipulating booster is a hydraulic position servomechanism with negative The  helicopter‐manipulating  booster  is  a  hydraulic  position  servomechanism  with  negative  feedback. It is composed of three basic parts, a dispensing mechanism (slide valve), an actuator feedback.  It  is  composed  of  three  basic  parts,  a  dispensing  mechanism  (slide  valve),  an  actuator  (piston) and a feedback mechanism (input rocker lever). In this valve-controlled hydraulic cylinder (piston) and a feedback mechanism (input rocker lever). In this valve‐controlled hydraulic cylinder  system, the displacement of the piston is controlled by the mechanical input of the rocker lever. system, the displacement of the piston is controlled by the mechanical input of the rocker lever. The  The electro-hydraulic loading system is used to evaluate the control performance with the load of the electro‐hydraulic loading system is  used  to  evaluate the control performance with the load of the  booster on on  the the  ground, and and  its schematic diagram is depicted in Figure 1. The left block is ablock  low-power booster  ground,  its  schematic  diagram  is  depicted  in  Figure  1.  The  left  is  a  electro-hydraulic position servo system, which is used to simulate the operating action on the control low‐power electro‐hydraulic position servo system, which  is  used to simulate the operating action  rodon the control rod by the driver’s hand. The control rod is linked to the rocker lever of the booster by  by the driver’s hand. The control rod is linked to the rocker lever of the booster by the linkage mechanism, and the rod’s action is reflected on the booster piston rod by amplifying the manipulating the linkage mechanism, and the rod’s action is reflected on the booster piston rod by amplifying the  force. The right block is an electro-hydraulic loading system of the valve-controlled symmetrical manipulating force. The right block is an electro‐hydraulic  loading system of the  valve‐controlled  hydraulic cylinder. The loading hydraulic cylinder piston rod is concatenated the booster piston rod symmetrical  hydraulic  cylinder.  The  loading  hydraulic  cylinder  piston  rod  to is  concatenated  to  the  bybooster piston rod by a force sensor. The load spectrum is exerted on the booster when the loading  a force sensor. The load spectrum is exerted on the booster when the loading hydraulic cylinder xM   and  of xL the hydraulic  cylinder  follows  the  motion xM of and the xLbooster.  Symbols    represent  the  follows the motion of the booster. Symbols represent the displacement booster piston and the loading hydraulic cylinder piston, respectively. displacement of the booster piston and the loading hydraulic cylinder piston, respectively. 

xM

xL  

Figure 1.  Schematic diagram  of the electro‐hydraulic loading system.  1. Booster; 2. Force sensor; 3.  Figure 1. Schematic diagram of the electro-hydraulic loading system. 1. Booster; 2. Force sensor; Loading servo valve; 4. Loading cylinder; 5. Control rod; 6. Linkage mechanism; 7. Motion cylinder;  3. Loading servo valve; 4. Loading cylinder; 5. Control rod; 6. Linkage mechanism; 7. Motion cylinder; 8. Motion servo valve.  8. Motion servo valve.

The studied electro‐hydraulic loading system is characterized by high‐frequency dynamic load  The studied electro-hydraulic loading system is characterized by high-frequency dynamic load superimposed on a large static load. The motion spectrum is compiled in sinusoidal form to simulate  superimposed onactions  a largeon  static load. The motion spectrum compiled frequency  in sinusoidal to simulate the  operating  the  control  rod  by  hand,  and  its ismaximum  is  2 form Hz.  The  load  spectrum is compiled to simulate the  aerodynamic load in the form of a sinusoidal dynamic load  superimposed on a large static load, and the frequency range of the dynamic load is generally from 

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the operating actions on the control rod by hand, and its maximum frequency is 2 Hz. The load to simulate the aerodynamic load in the form of a sinusoidal dynamic4 of 27  load superimposed on a large static load, and the frequency range of the dynamic load is generally from 30 to 80 Hz. That is different from the loading system of the fixed‐wing aircraft actuator, where the  30 to 80 Hz. That is different from the loading system of the fixed-wing aircraft actuator, where the frequency of the load signal is equal to that of the actuator position signal. In the loading system of  frequency of the load signal is equal to that of the actuator position signal. In the loading system of the the helicopter booster, the frequency of the load spectrum is much greater than that  of the booster  helicopter booster, the frequency of the load spectrum is much greater than that of the booster motion, motion, which results in the design of the controller of the loading system of the helicopter booster  which results in the design of the controller of the loading system of the helicopter booster being more being more difficult than that of a fixed‐wing aircraft actuator.  difficult than that of a fixed-wing aircraft actuator. Appl. Sci. 2016, 6, 98  spectrum is compiled

2.2. System Model and Analysis  2.2. System Model and Analysis In this paper, we do not discuss the issue of the manipulation simulation of the booster. Hence,  In this paper, we do not discuss the issue of the manipulation simulation of the booster. Hence, from the above descriptions of the manipulation simulation process, and to facilitate the study, the  from the above descriptions of the manipulation simulation process, and to facilitate the study, the booster can be regarded as an electromagnetic valve controlled a hydraulic cylinder that is driven by  booster can be regarded as an electromagnetic valve controlled a hydraulic cylinder that is driven by the electromagnetic valve input. The studied equivalent schematic diagram is shown in Figure 2.  the electromagnetic valve input. The studied equivalent schematic diagram is shown in Figure 2.

xM

xL

FL

Ks i v1

iv

xv

ps1

ps

 

Figure 2. The equivalent schematic diagram for modeling.  Figure 2. The equivalent schematic diagram for modeling.

The  The mathematical  mathematical model  model of  of the  the electro‐hydraulic  electro-hydraulic loading  loading system  system can  can be  be set  set up  up by  by means  means of  of theoretical  analysis  or  experiment  identification.  The  theoretical model  plays an  important  in  theoretical analysis or experiment identification. The theoretical model plays an important rolerole  in the the design of controllers and the selection of system parameters.  design of controllers and the selection of system parameters. As shown in Figure 2, the load force  (N) applied to the booster can be measured by the load  As shown in Figure 2, the load force FFL  (N) applied to the booster can be measured by the load L

  (N/m), the elastic stiffness of the force sensor, it can be expressed as  cell, and in consideration of  cell, and in consideration of KKs s(N/m), the elastic stiffness of the force sensor, it can be expressed as

( xLL ´  xxMM).q.  FLFL“KKsspx

(1)  (1)

The load flow equation of the loading valve with ideal zero opening can be written as  The load flow equation of the loading valve with ideal zero opening can be written as

qL  Cd wxvb( ps  sgn( xv ) pL )/ ,   (2)  qL “ Cd wxv pps ´ sgnpxv qpL q{ρ, (2) where  xv   is the spool displacement (m);  Cd   is the discharge coefficient;  w   is the area gradient of  where xv is the spool displacement (m); Cd is the3discharge coefficient; w is the area 2gradient of the );  ps   is the supply pressure (N/m );  pL   and  qL   the valve orifices (m);     is the  oil density (kg/m valve orifices (m); ρ is the oil density (kg/m3 ); ps is the supply pressure (N/m2 ); pL and qL denote the 2 denote the load pressure (N/m ) and the load flow (m3/s), respectively.  load pressure (N/m2 ) and the load flow (m3 /s), respectively. In  order  to  utilize  linear  system  theory  for  dynamic  analysis  on  the  hydraulic  power  In order to utilize linear system theory for dynamic analysis on the hydraulic power mechanism, mechanism, to linearize Equation (2) yields the linearized load flowrate equation as  to linearize Equation (2) yields the linearized load flowrate equation as qL  Kq xv  K c pL ,   (3)  q L “ Kq x v ´ Kc p L , (3) 2 5 where  K q   and  Kc   are  the  linearized flow  gain (m /s)  and  flow‐pressure  coefficient (m /(N∙s))  of  where Kq and Kc are the linearized flow gain (m2 /s) and flow-pressure coefficient (m5 /(N¨ s)) a K q CC w (pp pss´ppL L)/q{ρ    and  the  one  operating operating point, point,  respectively,  dw of theservo  servovalve  valve around  around one respectively, Kq “ and d a ˘ Kcc “ wxv /(2 p ) q. ρThe  . The values change with the variation of the operating v {p2(pspp q andKK  CCddwx  sp´ Kq K values  of  of   and  c  c change  with  the  variation  of  the  operating  L )L points. When the valve opening is very small, the flowrate‐pressure characteristics are  determined  by the leakage characteristics between the sleeve and the spool of the valve, and then  Kc   should be  calculated by  K c  rc2 w/(32)   according to the laminar flow formula.  The flowrate continuity equation of the hydraulic power mechanism can be expressed as 

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5 of 27  valve opening is very small, the flowrate-pressure characteristics are determined by the leakage characteristics between the sleeve and the spool of the valve, and then Kc should be calculated by Kc “ πrc2 w{ p32µq according todxthe laminar V dpL flow formula. qL  AL L  L  Csl pL ,   (4)  The flowrate continuity equation of thedthydraulic 4e dtpower mechanism can be expressed as

VL dp 2 dx 2 L fluid  where  AL   is  the  piston  effective  area(m qL “ AL);  Le  `is  the  ` bulk  Csl pL ,modulus  (N/m );  Csl   is  the  total  (4) dt 5 4βe dt leakage coefficient of the hydraulic cylinder (m /(N∙s)); and  VL   represents the total control volume  3).  where AL is the piston effective area(m2 ); βe is the fluid bulk modulus (N/m2 ); Csl is the total leakage of the loading system (m coefficient of the hydraulic cylinder (m5 /(N¨ s)); and VL represents the total control volume of the The force equilibrium equation for the loading hydraulic cylinder piston is  loading system (m3 ). d 2 xL dx The force equilibrium equationAfor the loading piston is hydraulic BL L  Fcylinder (5)  L pL  mL L,   2 dt dt d2 x dx AL pL “ mL 2L ` BL L ` FL , (5) where  mL   is the equivalent load mass (kg) including the mass of the piston, rods, force sensor and  dt dt the load; and  BL   represents the equivalent viscous damping coefficient (N∙s/m).  where mL is the equivalent load mass (kg) including the mass of the piston, rods, force sensor and the Combining and solving Equations (1) and (3)–(5), the transfer function of the output load force  load; and BL represents the equivalent viscous damping coefficient (N¨ s/m). FL ,  with  xM   of load the  valve  displacement    as (3)–(5), the  input  and  the function displacement  the  force loaded  Combining andspool  solving Equations (1)xvand the transfer of the output FL , booster as the disturbance, can be derived as follows  with the valve spool displacement xv as the input and the displacement xM of the loaded booster as the disturbance, can be derived as follows N (0) N ( s) FL ( s )  Gp ( s ) X v ( s )  Gd ( s ) sX M ( s )  U X v ( s )  D sX M ( s ),   (6)  PDN( s )p0q PD N ( s ) psq U D FL psq “ Gp psqXv psq ´ Gd psqsXM psq “ Xv psq ´ sXM psq, (6) P psq PD2 psq VL mL 2  VL D BL  AL  BL  s  ˆmL  ;  N U (0)  AL K q /K tm ;  where  Ktm  Kc  Csl ;  N D  s    s˙ A2 V 4L emKLtm 2  4 V KB K s ` mL ` e Ltm L s ` BL tm` L ; NU p0q “ AL Kq {Ktm ; where Ktm “ Kc ` Csl ; ND psq “ 4β K K¸tm ˜L2 4βeVKLtm   mL ˆ VL BL  e2 tm ˙ VL mL BL A 3 2 X ( PD  s   s   s    s  1 ;  AL  V m m V B  B VL v s)   and  X M (s)   denote   4e K tmLK s L s3 K`s 4Le K K tm K sL `4e K tm PD psq “ `tm K s L L  K ss2 ` ` s ` 1; Xv psq and XM psq  4βe Ktm Ks Ks 4βe Ktm Ks Ks Ktm Ks 4βe Ktm xv   and  the Laplace transform of  , respectively.  denote the Laplace transform of xvxMand xM , respectively.   is related  It can be noticed from Equation (6) that, besides the valve spool displacement  It can be noticed from Equation (6) that, besides the valve spool displacement xxvv, , FFLLis related to thethe  booster displacement xM asxMwell. latter disturbance exactly results in results  the extraneous to  booster  displacement    as  The well.  The motion latter  motion  disturbance  exactly  in  the  force in the loading system. extraneous force in the loading system.  The spool movement dynamics of the loading valve cannot be ignored within the bandwidth of The spool movement dynamics of the loading valve cannot be ignored within the bandwidth of  our studied loading system. And a second-order dynamic equation is therefore utilized to describe the our studied loading system. And a second‐order dynamic equation is therefore utilized to describe  relationship between the coil current input iv (A) the spool displacement xv , asxit , as it can well  can well reflect iv  and the relationship between the coil current input  (A) and the spool displacement  v the valve spool dynamics at high frequencies, and its transfer function can be expressed as reflect the valve spool dynamics at high frequencies, and its transfer function can be expressed as 

GG “ v psq v (s) 

2 2 XXvvpsq KK (s) vω v v v “  22  2 2 IIvvpsq v s ` ωv vω ( s) ss `22ζ v v s  v

(7) (7)

3 /(A¨ s)); where Iv psq Laplace transform of of  iv ; K valve spool spool  position gain (m ωv andvζv  (s)  isis the iv v;  isKthe the  Laplace  transform  the  valve  position  gain  (m3/(A∙s));  where  v   is  are the frequency (rad/s) and the damping ratio of the loading valve, respectively.  v   natural and  are the natural frequency (rad/s) and the damping ratio of the loading valve, respectively.  Therefore, a simplified block diagram of the loading system can be drawn in Figure 3, where Fd is Therefore, a simplified block  diagram of the loading system can be drawn in Figure 3, where  output disturbance force from the motion of the loaded plant and Fd “ Gd psqsXM . Fthe d   is the output disturbance force from the motion of the loaded plant and  Fd  Gd (s) sX M . 

xM

Gd ( s ) iv

Gv ( s )

xv

Gp ( s )

Fd

FL  

Figure 3. Simplified block diagram of the loading system. Figure 3. Simplified block diagram of the loading system. 

2.3. Equivalent Linear Model Set 

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2.3. Equivalent Linear Model Set Normally, we linearize the flow quantity equation of the valve around the neutral position, because the general hydraulic servo system is often working around that, and Kq and Kc are obtained as fixed constants. However, for the hydraulic loading system with strong position interference, its operating point varies over a large range. If the loading system is just described by a linear plant with linear parameters around one operating point, it is only possible to approximate the system dynamics over a limited region of operation. Moreover, parameter uncertainty in some hydraulic systems is inevitable and this can degrade the closed-loop performance. In particular, with the requirement of high closed-loop bandwidth (80 Hz), the inaccurately described system leads to unreliable design. Therefore, any single linear time invariant (LTI) model cannot describe the whole system’s dynamics, and it is necessary to establish a set of equivalent LTI models that describe the dynamics of the nonlinear hydraulic system over the whole envelope of operation. From the design point of view, we believe that the transfer function Gv psq can accurately reflect the dynamic characteristics of the servo valve, and some structural parameters, such as VL , AL , Ks , etc., can be precisely measured and kept constant. However, the equivalent load mass mL , which is described by the lumped mass method, is difficult to measure precisely, and the fluid bulk modulus βe fluctuates with fluid temperature and mixed air. According to the maximum load demand of the booster with some appropriate margin (from ´11 to 22 kN), the initial boundaries of Kq and Kc can be calculated. The electro-hydraulic loading system generally uses the servo valve with zero opening or positive opening, and even the zero-opening servo valve generally also has a very small positive opening (under-lapped). The positive opening and the variation of oil supply pressure ps also result in the perturbations of Kq and Kc . A lumped perturbation range can be obtained by integrating these perturbations into the initial boundaries of Kq and Kc . Then, taking into account the leakage of the whole system can get the boundary of Ktm . Table 1 lists all the measured and calculated values of the parameters of the loading system. Table 1. Parameters of the loading system. Parameter

Value

Unit

Supply pressure ps Piston effective area AL Total control volume VL Equivalent load mass mL Fluid bulk modulus βe Viscous damping coefficient BL Force sensor stiffness Ks Valve flow gain Kq Total flow-pressure coefficient Ktm Valve gain Kv Valve natural frequency ωv Valve damping ratio ζv Feedback gain Ka

18–21 24.63 4.5 8–12 700–1100 800 100 2.92–3.85 7 ˆ 10´2 –7.5 ˆ 10´2 0.3/20 200 0.7 4 ˆ 10´4

MPa cm2 cm3 kg MPa N¨ s/m kN/mm m2 /s cm5 /(N¨ s) mm/mA Hz mA/N

From the model derived above, the loading system is a fifth-order system with motion disturbance and variable parameters. The denominator polynomial PD psq of GP psq can be expressed as follows ˆ PD psq “

¸ ˙˜ 2 s s 2ξh `1 ` s`1 ωL ωh ω2h

(8)

“ where ωL is the cutoff frequency of the inertial term and ωL “ Ktm Kh Ks { A2L pKh ` Ks qs ; ωh and ξh are the second-order oscillation frequency and the damping ratio of the force loop, b respectively; a 2 ωh “ ω0 pKh ` Ks q{Kh ; Kh “ 4βe AL {VL is the liquid spring stiffness, ω0 “ 4βe A2L { pVL mL q

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where  L   is the cutoff frequency of the inertial term and  L  K tm K h Ks /[AL2 ( K h  Ks )] ;  h   and  h   are  the  second‐order  oscillation  frequency  and  the  damping  ratio  of  the  force  loop,  respectively;  Appl. Sci. 2016, 6, 98

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h  0 ( K h  Ks )/K h ;  K h  4e AL2 /VL   is the liquid spring stiffness,  0  4e AL2 /(VL mL )   represents  the  natural  angular  frequency  of  the  hydraulic  cylinder;  h   can  be  calculated,  but  the  symbolic  represents the natural angular frequency of the hydraulic cylinder; ξh can be calculated, but the representation is difficult to obtain.  symbolic representation is difficult to obtain. After  some  calculation,  the  loading  system  P(s)   can  be  represented  as  an  equivalent  linear  After some calculation, the loading system Ppsq can be represented as an equivalent linear model model set  P0 (s)   having four uncertain parameters as  set P0 psq having four uncertain parameters as $   , ’ / ’  / ’ / ’  / K op Gv ( s ) &  . Kop Gv psq ( ) ( )   P s P s   (9)    ˜ ¸ 2 Ppsq P P0 psq 0“ ˆ  s (9) ˙  /   s 2 2 h ’  ’ s  1  2s  2ξs h 1 / ’ / ’ / ` 1  h 2 ` h ω s ` % ω  L  1 ω L h h

109 ] .  In  where  L [31.62, 46.03] ;  h [3387, 4464] ;  h  [0.012,0.015] ;  and  K op  [9.56 108 ,1.354 8 where ωL P r31.62, 46.03s; ωh P r3387, 4464s; ξh P r0.012, 0.015s; and Kop P r9.56 ˆ 10 , 1.354 ˆ 109 s. light of the statistics of the hydraulic servo system, the damping ratio of the hydraulic servo system is  In light of the statistics of the hydraulic servo system, the damping ratio of the hydraulic servo system   0.1 generally more than 0.1, and therefore we take  is generally more than 0.1, and therefore we takeh ξh “ . 0.1. From  From the  the frequency  frequency characteristics  characteristics of  of the  the equivalent  equivalent model  model set  set in  in Figure  Figure 4,  4, the  the dynamic  dynamic characteristics of the loading system can be described by the linearized equivalent model set, but this  characteristics of the loading system can be described by the linearized equivalent model set, but equivalent  method  introduces  a  significant  amount  of of dynamic  uncertainty  this equivalent method introduces a significant amount dynamic uncertaintyin  inthe  thelow‐frequency  low-frequency region, which is smaller in the actual plant.  region, which is smaller in the actual plant.

Magnitude (dB)

150

100

50

0 0

Phase (deg)

-90 -180 -270 -360 -450 0 10

1

10

2

10 Frequency (rad/s)

3

10

4

10

 

Figure 4. Frequency characteristics of equivalent model set.  Figure 4. Frequency characteristics of equivalent model set.

2.4. Disturbance Force Analysis  2.4. Disturbance Force Analysis It is known from Equation (6) that the output loading force  consists of two parts: one part  It is known from Equation (6) that the output loading force FFLL  consists of two parts: one part is controlled  by regulating the spool displacement  , and the other  Fd   results  Fvv  ) X v v( psq s )   is “GGp p( spsqX controlled by regulating the spool displacement xvx, and the other Fd results from v the displacement xM of the plant. This disturbance force Fd is inevitably produced because of of the loaded plant. This disturbance force  from  the displacement  xM  loaded Fd  . is inevitably produced  the motion of the loaded plant. The amount of F depends on the velocity x of the booster, and is M d because of the motion of the loaded plant. The amount of  Fd   depends on the velocity  x M   of the  related to the transfer function Gd psq. According to the given parameters in Table 1 and computing the booster, and is related to the transfer function  G d ( s ) . According to the given parameters in Table 1  relative terms in Equation (6), the frequency characteristic curves of Fd for each equivalent linear model and computing the relative terms in Equation (6), the frequency characteristic curves of  Fd   for each  in P0 psq are acquired and are shown in Figure 5. As seen from Figure 5, the disturbance force Fd has equivalent linear model in  (s)   are acquired and are shown in Figure 5. As seen from Figure 5, the  differential characteristics atP0low frequencies and large magnitude (in dB).The magnitude of Fd rises disturbance  force    has  characteristics  at  low  frequencies  magnitude  (in  Fd dB with the slope of +20 as differential  the frequency increases at low frequencies, tendand  to belarge  saturated and stable in the middle frequency and sharply rise in high frequency region. In practice, the frequency dB).The  magnitude  of  Fband, d   rises  with  the  slope  of  +20  dB  as  the  frequency  increases  at  low  of the position interference will not be high. Meanwhile, at low frequencies (less than 30 rad/s), the

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Appl. Sci. 2016, 6, 98 8 of 26 frequencies, tend to be saturated and stable in the middle frequency band, and sharply rise in high  frequency region. In practice, the frequency of the position interference will not be high. Meanwhile,  at low frequencies (less than 30 rad/s), the amount of  Fd(i)   varies a little, about 1 dB for the models in  piq amount of Fd varies a little, about 1 dB for the models in P0 psq, which means model perturbation has P0 (s) , which means model perturbation has a small effect on the disturbance force of our interest.  a small effect on the disturbance force of our interest. Disturbance Force Frequency Characteristic 200

Magnitude (N in dB)

180 160 140 120 F(i) d 100 0

10

1

10

2

10 Frequency (rad/s)

3

10

4

10

 

Figure 5. Frequency characteristic set of the extraneous force with uncertainty.  Figure 5. Frequency characteristic set of the extraneous force with uncertainty. . Despite  this disturbance disturbance force force    is  helpful  when  the  direction  FL   is  opposite.  Despite this Fd Fisd helpful when the direction of xM of  andx MFL  and  is opposite. However, However, generally, such a disturbance force will mostly increase the force error between the input  generally, such a disturbance force will mostly increase the force error between the input and output, and output, and it affects force‐tracking accuracy. From the perspective of system control, this force  and it affects force-tracking accuracy. From the perspective of system control, this force error affects error  affects  the  closed‐loop  bandwidth  of  the and loading  system,  and ofwith  the  increase  of  that the  the closed-loop bandwidth of the loading system, with the increase the disturbance force, disturbance force,  that  bandwidth  ideal loading  bandwidth is seriously narrowed.is seriously narrowed. It is difficult to achieve an  It is difficult to achieve an ideal loading performance before performance  before  this has disturbance  force  has  been  effectively  the  problem  of  this disturbance force been effectively suppressed. Thus, suppressed.  the problemThus,  of disturbance force disturbance  force  suppression  should  be loading solved system; firstly otherwise, in  the  loading  system;  otherwise,  the  suppression should be solved firstly in the the closed-loop bandwidth and closed‐loop bandwidth and the force‐tracking accuracy cannot be guaranteed.  the force-tracking accuracy cannot be guaranteed. Generally,  the motion motion amplitude amplitude and and  frequency  booster  measurable  bounded;  Generally, the frequency of of  thethe  booster are are  measurable and and  bounded; thus, thus,  the  magnitude  and  frequency  of  be   must  be and bounded  and  measurable.  In maneuvering a  helicopter  Fd bounded the magnitude and frequency of Fd must measurable. In a helicopter system, the frequency xM frequency  is low, so Fdof can seen as aso  low-frequency low,  seen  as  a  low‐frequency  maneuvering  system, ofthe  x Mbe  is  Fd   can  be disturbance.

disturbance.  3. Design of Hybrid Control Scheme 3. Design of Hybrid Control Scheme  From the analysis above, strong disturbance and large parameter perturbation exist in the controlled plant. Despite that, other uncertainties such as friction and the gap in mechanical connection, From  the  analysis  above,  strong  disturbance  and  large  parameter  perturbation  exist  in  the  which are unknown disturbances, also exist. Therefore, designing a robust control method for the controlled  plant.  Despite  that,  other  uncertainties  such  as  friction  and  the  gap  in  mechanical  controlled plant is needed. connection,  which  are  unknown  disturbances,  also  exist.  Therefore,  designing  a  robust  control  Since robust control relates the amount of feedback at each frequency to the amount of uncertainty method for the controlled plant is needed.  in the plant dynamics, the equivalent linear method forces the magnitude of the loop transmission Since  robust  control  relates  the  amount  of  feedback  at  each  frequency  to  the  amount  of  to be unnecessarily large in the low-frequency range. If we take the robust approach directly, the uncertainty in  the  plant  dynamics,  the equivalent linear  method forces  the  magnitude  of  the  loop  result is an over-designed and conservative control system. Because it uses much more feedback gain transmission to be unnecessarily large in the low‐frequency range. If we take  the robust approach  than is actually required to solve the robust control problem, it does not even satisfy the performance directly, the result is an over‐designed and conservative control system. Because it uses much more  requirements of the system. Thus, in order to effectively solve the control and synthesis for a loading feedback gain than is actually required to solve the robust control problem, it does not even satisfy  system with large perturbation and strong motion disturbance, a hybrid control scheme is presented. the  performance  requirements  of  the  system.  Thus,  in  order  to  effectively  solve  the  control  and  synthesis  for Feed-Forward a  loading  system  with  large  perturbation  and  strong  motion  disturbance,  a  hybrid  3.1. Constant Compensator control scheme is presented.  As seen from Figures 3 and 5 Fd is a measureable and bounded disturbance with large magnitude. From the perspective of control, using the conventional feed-forward control can suppress this 3.1. Constant Feed‐Forward Compensator  disturbance, because the feed-forward control can compensate for the effect of this disturbance on As  seen  from  Figures  3  and  5,  Fd   is  a  measureable  and  bounded  disturbance  with  large  system output before adverse effects. If the feed-forward regulator compensates the disturbance just magnitude.  From  the  perspective  control,  using  the  conventional  feed‐forward  control  can  right, the controlled variable will notof produce deviation. suppress  this  disturbance,  because  the structure feed‐forward  control  can  compensate  for traditional the  effect  of  this  A control strategy based on the invariance principle is the most one to compensate for this disturbance force because it does not change the structure of the closed loop, and

A  control  strategy  based  on  the  structure  invariance  principle  is  the  most  traditional  one  to  compensate for this disturbance force because it does not change the structure of the closed loop,  and  it  can  be  realized  by  software  easily.  According  to  the  structure  invariance  principle,  if  we  choose  the  velocity  x M   as  the  input  (the  observed  motion  disturbance)  of  the  compensator  and  make  Appl. Sci. 2016, 6, 98

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N D (s) U c ( s )  Gcom ( s ) X M ( s )  X M ( s )   (10)  N (0) G ( s ) U v it can be realized by software easily. According to the structure invariance principle, if we choose the . velocity xM as the input observed disturbance) of thethat  compensator and make    will be canceled, and    caused  by  xmotion means the extraneous force is  the disturbance  force  F (the d

M

. . suppressed  completely.  However,  the  robustness  of Nthis  D psq compensation  is  bad  for  system  X M psq (10) Uc psq “ Gcom psqX M psq “ NU p0qG nonlinearity  and  time‐varying  parameters.  In  general,  the v psq velocity  of  the  booster  cannot  be  . measured directly, and the rapidity and accuracy of the velocity obtained by the differential method  the disturbance force Fd caused by xM will be canceled, and that means the extraneous force is or  velocity  estimation  seriously  affect  the  restraining  ability.  Moreover,  G com ( s )   is  a  non‐proper  suppressed completely. However, the robustness of this compensation is bad for system nonlinearity transfer function, and it is difficult in physical implementation.  and time-varying parameters. In general, the velocity of the booster cannot be measured directly, and In  this  case,  the  frequency  less  than by3 the Hz, differential and  then  method be  approximately  x M   is obtained G v ( s )   can  the rapidity and accuracy of the of  velocity or velocity estimation equal to  v (0) . It is well known that most of the disturbance force is related to the velocity of the  seriouslyGaffect the restraining ability. Moreover, Gcom psq is a non-proper transfer function, and it is motion  in  the  low‐frequency  range.  Then,  a  compensation  term  can  be  determined  difficultdisturbance  in physical implementation. . when ignoring the high‐order factors in  N D (than s )   to eliminate the main part of the disturbance force  In this case, the frequency of xM is less 3 Hz, and then Gv psq can be approximately equal to G p0q. It is well known that most of the disturbance force is related to the velocity of the motion as follows  v disturbance in the low-frequency range. Then, a compensation term can be determined when ignoring N D (0)   disturbance force as follows (11)  ( s ) main part Gcom the the high-order factors in ND psq to eliminate of the N U (0)Gv (0)

ND p0q There are time‐varying parameters in  N D (0) N (0) , and we take their middle values to  Gcom psq “   and  (11) NU p0qGvUp0q obtain the constant compensation term  Gcom (0)  0.0486 .  There are time-varying parameters in ND p0q and NU p0q, andP0we take their middle values to obtain The characteristics of the disturbance force of models in  (s) , which are compensated by the  the constant compensation term G p0q “ 0.0486. com constant  compensator,  are  depicted  in  Figure  6.  The  velocity  x M   is  acquired  by  the  direct  The characteristics of the disturbance force of models in P0 psq, which are compensated by the differential  method.  In  Figure  6,  F d'(i)   represents  the  . remaining  disturbance  force  after  constant compensator, are depicted in Figure 6. The velocity xM is acquired by the direct differential (i) compensation, and  Fd   1prepresents the original disturbance force.  iq piq method. In Figure 6, Fd represents the remaining disturbance force after compensation, and Fd Although the restraining  effect  on  the  disturbance force  of models in  P0 (s)   differs within the  represents the original disturbance force. concerned frequency range, the constant compensator at least attenuates the disturbance force 16 dB  Although the restraining effect on the disturbance force of models in P0 psq differs within the in magnitude; thus, at least 84% of the disturbance force is eliminated. However, the feed‐forward  concerned frequency range, the constant compensator at least attenuates the disturbance force 16 dB compensator is sensitive to phase and the accuracy of the velocity signal. The restraining effect will  in magnitude; thus, at least 84% of the disturbance force is eliminated. However, the feed-forward degrade because it is difficult to get an ideal velocity signal in practical implementation. Therefore,  compensator is sensitive to phase and the accuracy of the velocity signal. The restraining effect will the remaining extraneous force, after being compensated, will not be small enough, it will still affect  degrade because it is difficult to get an ideal velocity signal in practical implementation. Therefore, the the  force‐tracking  performance,  is  necessary  will to  restrain  the  remaining  extraneous  force  remaining extraneous force, after and  beingit compensated, not be small enough, it will still affect the further.  force-tracking performance, and it is necessary to restrain the remaining extraneous force further. Disturbance Force Frequency Characteristic

Magnitude (N in dB)

200

150

(i)

100

Fd

F'(i) d 50 0 10

1

10

2

10 Frequency (rad/s)

3

10

4

10

 

Figure 6. Disturbance force after constant compensation.

3.2. DOB-based Compensator The remaining extraneous force, after been compensated by the constant velocity compensation, becomes an immeasurable disturbance with relatively small magnitude. From the perspective of control, the immeasurable disturbance can be compensated by an observer.

3.2. DOB‐based Compensator  The  remaining  extraneous  force,  after  been  compensated  by  the  constant  velocity  compensation,  becomes  an  immeasurable  disturbance  with  relatively  small  magnitude.  From  the  perspective of control, the immeasurable disturbance can be compensated by an observer.  Appl. Sci. 2016, 6, 98 10 of 26 If controllers of the loading system are designed based on a double‐loop cascade composition  control strategy, two problems of disturbance rejection (including the remaining extraneous force)  If controllers of the loading system are designed based on a double-loop cascade composition and  force  tracking  can  be  solved  separately.  The  inner  loop  controller  is  utilized  to  eliminate  the  control strategy, two problems of disturbance rejection (including the remaining extraneous force) and remaining extraneous force and other disturbances, enhance system robustness, and make the actual  force tracking can be solved separately. The inner loop controller is utilized to eliminate the remaining plant behave as a nominal model approximately within certain operating bandwidth.  The nominal  extraneous force and other disturbances, enhance system robustness, and make the actual plant behave model–based  outer  loop  controller  is  used  to  realize  the  desired  force‐tracking  performance.  It  is  as a nominal model approximately within certain operating bandwidth. The nominal model–based more reasonable to estimate the disturbance on the control channel than to estimate the disturbance  outer loop controller is used to realize the desired force-tracking performance. It is more reasonable to itself because we can use the control input to improve the disturbance rejection performance. Then  estimate the disturbance on the control channel than to estimate the disturbance itself because we can the disturbance observer (DOB) method is introduced in Figure 7, where symbol  d 0   represents the  use the control input to improve the disturbance rejection performance. Then the disturbance observer equivalent  disturbance  caused  by  un‐modeled  dynamics  and  unknown  disturbances  including  (DOB) method is introduced in Figure 7, where symbol d0 represents the equivalent disturbance caused friction.  by un-modeled dynamics and unknown disturbances including friction.

Gcom ( s )

xM sGd ( s )

uvc

uc

s

d0

ui

P( s)

udo

Fd

FL

Pn-1 ( s) Q( s)   Figure 7. Disturbance observer (DOB) structure for the compensated system.  Figure 7. Disturbance observer (DOB) structure for the compensated system.

The  uncertainty  of  the  system  can  be  treated  as  a  multiplicative  perturbation  of  the  nominal  The uncertainty of the system can be treated as a multiplicative perturbation of the nominal system, and it is given as  system, and it is given as P ( s“ ) PPnnpsqp1 ( s )(1 `∆LL( spsqq ))   (12)  Ppsq (12) where ∆L Lis  is an allowable multiplicative uncertainty.  where an allowable multiplicative uncertainty. The DOB a lumped disturbance, which which  includesincludes  the effectsthe  dueeffects  to model uncertainties The  DOB estimates estimates  a  lumped  disturbance,  due  to  model  as well as the external from the difference between the input-output relation of an actual uncertainties  as  well  disturbances, as  the  external  disturbances,  from  the  difference  between  the  input‐output  plant Ppsq nominal model Thenominal  amountmodel  of the equivalent estimated relation  of and an  its actual  plant  amount  of  dthe  P ( s )  Pand  Pn (s) .  The  disturbance n psq. its  0 isequivalent  ´1 psq, and then the equivalent compensation 1 through the inverse nominal model P is introduced into disturbance  d 0   is  estimated  through  n the inverse  nominal  model  Pn ( s ) ,  and  then  the  equivalent  the control through a low-pass filter Qpsq, i.e., the cancellation input u is fed back into the control do Q ( s ) ,  i.e.,  the  cancellation  compensation  is  introduced  into  the  control  through  a  low‐pass  filter  input, and thus a complete restraining of the equivalent disturbance is achieved. input  udo   is  fed  back  into  the  control  input,  and  thus  a  complete  restraining  of  the  equivalent  Defining d “ puvc ` d0 qP ´ Fd , the transfer functions from each relevant input to output FL can be disturbance is achieved.  written as Defining  d  (uvc  d 0 ) P  Fd , the transfer functions from each relevant input to output  FL   can  F PPn be written as  (13) GFL uc “ L “ uc QpP ´ Pn q ` Pn GFL d “ GFL x. M “

FL Pn p1 ´ Qq “ d QpP ´ Pn q ` Pn

FL pGcom p0qP ´ Gd qPn p1 ´ Qq “ QpP ´ Pn q ` Pn xM .

FL “ GFL uc uc ` GFL d d

(14) (15) (16)

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.

In Equation (15), if Qpsq « 1 in the low-frequency range r0, 2s (Hz), the output from xM to FL approximately equals zero, which means the disturbance force is well suppressed. . It is noticed that there is the same multiplicative factor 1 ´ Qpsq from d and xM to the output force FL , which means that the disturbance force can be regarded as the output resulting from the external disturbances. So the disturbance force can be regarded as a low-frequency disturbance. The sensitivity function SDOB and the complementary sensitivity function TDOB of the DOB loop are derived as $ 1 ´ Qpsq ’ ’ ’ & SDOB “ 1 ` Qpsq∆ psq L (17) ’ p1 ` ∆L psqqQpsq ’ ’ % TDOB “ 1 ` Qpsq∆L psq From the above analysis, the design of the DOB in our hybrid suppression scheme coincides with the design of the general DOB. When Qpsq is designed, its cut-off frequency must be higher than the frequency of the disturbance force. At the same time, the frequency responses of sensitivity functions with the DOB should below enough to ensure sufficient disturbance attenuation besides the constraints of the traditional DOB, because the plant uncertainty is negligible in the low-frequency region. It is well known that the key of the DOB design is the design of Qpsq. Once the nominal model is chosen, the upper bound of the uncertainty against the nominal model is confirmed, and the upper limitation bandwidth of Qpsq is decided. The tracking performance is enhanced by increasing the order and the bandwidth of Qpsq, but the excessive order and bandwidth of Qpsq can lead to resultant destruction of the stability condition due to plant uncertainty in the high-frequency region. The design of Qpsq is a trade-off between disturbance suppression and robustness. If Qpsq is designed appropriately, the disturbance observer not only suppresses the rest disturbance, but also fulfills robustness. Apparently, solving the model mismatch in an overall frequency band through the DOB loop compensator is unrealistic. Therefore, the DOB-based inner loop compensator is designed mainly to solve the suppression problem of the extraneous force, other low-frequency external interference and model mismatch at low frequencies, as well as taking some proper consideration of the system model mismatch at middle and high frequencies. The model mismatch in the middle- and high-frequency range can be considered in the outer loop control. 3.3. Robust Performance Controller Equation (13) can be transformed into „ GFL uc psq “ Pn psq 1 `

1 ´ Qpsq ∆L psq 1 ` Qpsq∆L psq

 (18)

The controlled plant is changed because of the feedback of the DOB loop. However, from the relationship between uc to FL in Equation (18), it is obvious that the outer loop controlled plant is still a model with multiplicative perturbation, and its structure can be equivalent to a virtual controlled 1 ´ Qpsq plant Pn psqp1 ` ∆1 q, where Pn psq is the nominal model and ∆1 “ ∆L psq. 1 ` Qpsq∆L psq In our loading system with dynamic load superimposed on static load, the force signal to be tracked is in the sinusoidal form or the form of sine superposition offset; a feedback controller is therefore adequate for the outer loop. Therefore, a robust feedback controller based on the H8 theory is designed, and the structure of the outer loop is shown in Figure 8.

1  Q ( sL) controlled plant  Pn ( s)(1  1 ) , where  Pn (s)   is the nominal model and  1   L ( s ) .  In our loading system with dynamic load superimposed on static load, the force signal to be  1  Q( s ) L ( s ) tracked  is  in  the  sinusoidal  form  or  the  form  of  sine  superposition  offset;  a  feedback  controller  is  In our loading system with dynamic load superimposed on static load, the force signal to be  therefore  for  the  outer  loop.  a  robust  feedback  controller  based controller  on  the  H∞  tracked  is adequate  in  the  sinusoidal  form  or  the Therefore,  form  of  sine  superposition  offset;  a  feedback  is  theory is designed, and the structure of the outer loop is shown in Figure 8.  therefore  adequate  for  the  outer  loop.  Therefore,  a  robust  feedback  controller  based  on  the  H∞  Appl. Sci. 2016, 6, 98

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theory is designed, and the structure of the outer loop is shown in Figure 8. 

ur ur

Gc ( s ) Gc ( s )

uc uc

 Q1((s )  Q1((s )

Pn ( s ) Pn ( s )

FL FL

Ka Ka

 

Figure 8. Equivalent structure of the outer loop. 

 

Figure 8. Equivalent structure of the outer loop. 

Figure designed,  8. Equivalent structure the outer loop. of  the  outer  loop  feedback  If  the  Q( s)   is  appropriately  the  design ofspecifications 

controller  may  consider  the  disturbance  attenuation  requirement.  the  outer  loop  If  the Qpsq   appropriately is  appropriately  designed,  the  design  specifications  of Therefore,  the  outer  loop  feedback  Q( sis ) not  If the designed, the design specifications of the outer loop feedback controller controller    may  not  include  an  integrator.  From  the  small  gain  theorem,  the  sufficient  and  G ( s ) c controller  may  not  the attenuation disturbance requirement. attenuation  Therefore, requirement.  the  outer Gloop  may not consider theconsider  disturbance the Therefore,  outer loop controller c psq necessary condition for the stability and robustness of    is  gain  theorem,  the  sufficient  and  Gthe  )small  controller    may  not  include  From  G c ( s )an c ( sthe may not include integrator. From an  theintegrator.  small gain theorem, sufficient and necessary condition for necessary condition for the stability and robustness of  G c (Ps G )   is  the stability and robustness of Gc psq is 1Tou  1 and Tou  n c   (19)  1 P P ˇˇ ˇˇ Pn GGc n Gnc c ˇˇ ∆ 1 T ˇ ˇ   ă and Tou “ (19) ou (19)  1Tou  1 and Tou 1` Pnc Gc 1 Pn G 3.4. Hybrid Control Scheme  3.4. Hybrid Control Scheme 3.4. Hybrid Control Scheme  From  the  design  above,  the  hybrid  control  scheme  is  obtained  and  its  schematic  diagram  is  From the design above, the hybrid control scheme is obtained and its schematic diagram is shown shown in Figure 9.  From  in Figure 9.the  design  above,  the  hybrid  control  scheme  is  obtained  and  its  schematic  diagram  is  shown in Figure 9. 

s s

Gcom ( s ) Gcom ( s )

ur ur

u Gc ( s ) c u Gc ( s ) c

uvc uvc udo udo

ui ui

Q( s) Q( s)

P(s) P(s)

xM xM sGd ( s ) sGd ( sF)

d

Fd

FL FL

Pn-1 ( s ) Pn-1 ( s )

Ka Ka

 

Figure 9. Schematic diagram of the hybrid control scheme for the loading system.  Figure 9. Schematic diagram of the hybrid control scheme for the loading system.

 

Figure 9. Schematic diagram of the hybrid control scheme for the loading system. 

The  hybrid hybrid  control control  scheme scheme  consists consists  of of  aa  constant constant  velocity velocity  feed-forward feed‐forward  compensator compensator  and and  The double‐loop  cascade  composition  control  strategy.  The  constant  velocity  compensator  generates  The  hybrid  control  scheme  consists  of  a  constant  velocity  velocity feed‐forward  compensator  and  double-loop cascade composition control strategy. The constant compensator generates double‐loop  cascade  composition  control  strategy.  The  constant  velocity  compensator  generates  compensation input uvc to eliminate most of the extraneous force. The disturbance observer–based inner loop compensator generates compensation input udo to restrain disturbances besides the remaining extraneous force, and makes the actual plant tracking approximately a nominal model in a certain frequency range. In addition, the robust outer loop controller generates uc to achieve the desired force-tracking accuracy, and guarantees system robustness in the high-frequency region. Thus, the control law of the hybrid control scheme is obtained. The control input ui consists of three parts with different functions as follows ui “ uc ` uvc ´ udo

(20)

desired force‐tracking accuracy, and guarantees system robustness in the high‐frequency region.  Thus, the control law of the hybrid control scheme is obtained. The control input  ui   consists of  three parts with different functions as follows  ui  u c  u vc  u do  

(20) 

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Under the same model perturbation, the tracking performance of the robust controller that can  be achieved is limited. The direct robust method might achieve the same disturbance suppression  Under the same perturbation, the tracking performance of the robust that robust  can be ability  (in  dB)  with model the  hybrid  control  scheme.  However,  the  tracking  error controller of  the  direct  achieved The direct method might achieve same disturbance suppression ability method  is is limited. far  greater  than robust that  of  the  hybrid  control thescheme,  and  therefore  the  tracking  (in dB) with the hybrid control scheme. However, the tracking error of the direct robust method is far performance of the direct robust method is lower than that of the hybrid control scheme.  greater than that of the hybrid control scheme, and therefore the tracking performance of the direct 3.5. Design Method of Q(s)  robust method is lower than that of the hybrid control scheme. The  design  and  optimization  of  Q( s)   in  Figure  7  is  not  easy  when  the  requirements  of  3.5. Design Method of Q(s) performance  and  robustness  must  be  considered  together,  especially  when  there  are  some  The design and optimization of Qpsq in Figure 7 is not easy the requirements of performance performance demands. A systematic  method  in  dealing  with when this issue  was  proposed in [32].  The  and robustness must be considered together, especially when there are some performance demands. block diagram transformation for Figure 7 yields Figure 10a. In order to use robust control theory,  A systematic method in dealing with this issue was proposed in [32]. The block diagram transformation we take  for Figure 7 yields Figure 10a. In order to use robust control theory, we take P (s) K (s) Q ( s )  Pnn psqKpsq   (21)  1  Pn ( s ) K ( s ) Qpsq “ (21) 1 ` Pn psqKpsq where the internal‐loop compensator  K (s)   is stable. Then the structure of the hybrid suppression  where the internal-loop compensator Kpsq is stable. Then the structure of the hybrid suppression scheme  can  be  transformed  equivalently  to  a  structure  in  Figure  10b.  In  this  way,  the  design  of  scheme can be transformed equivalently to a structure in Figure 10b. In this way, the design of Qpsq Q( s )   can be transformed to the design of  K (s) .  can be transformed to the design of Kpsq.

uc

1 1  Q( s )

u0

(uvc ) ui

d

P( s )

FL

Q( s ) Pn ( s )

uc

Pn ( s)

K (s)

udo

u0

(uvc ) ui

d

P( s )

FL

  Figure 10. Equivalent structure of DOB: (a) from diagram transformation; (b) by definition.  Figure 10. Equivalent structure of DOB: (a) from diagram transformation; (b) by definition.

In Figure 10,  K (s)  isis the robust internal‐loop feedback controller. If  K (s)  isis designed for the  In Figure 10, Kpsq the robust internal-loop feedback controller. If Kpsq designed for the nominal model PnPpsq in order to satisfy a given performance and robustness criterion, the optimal Qpsq nominal model    in order to satisfy a given performance and robustness criterion, the optimal  ( s ) n of the DOB can be systematically designed by the optimal method under the given specific conditions, specific  Q( s )   of  the  DOB  can  be  systematically  designed  by  the  optimal  method  under  the  given  because the transfer function of the unity feedback system with P psq and Kpsq can determine in n conditions,  because  the  transfer  function  of  the  unity  feedback  system  with  Pn (s)   and  K (Qpsq s)   can  Equation (21). determine  Q( s)   in Equation (21).  The H8 mixed sensitivity method is utilized to derive the optimal robust internal loop compensator Kpsq, and then the optimal Qpsq can be derived by Equation (21). The sensitivity function SQ and the complementary sensitivity function TQ of Qpsq are obtained from Equation (22), respectively SQ “

1 Pn psqKpsq , TQ “ 1 ` Pn psqKpsq 1 ` Pn psqKpsq

(22)

Apparently, SQ “ 1 ´ Q and TQ “ Q. Then the H8 mixed sensitivity problem can be described as solving Kpsq to meet the minimum H8 norm ˇˇ ˇˇ ˇˇW p1 ´ Qqˇˇ ˇˇ ă 1 min ˇˇˇˇ 1 (23) Kstabilizing W3 Q ˇˇ8

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Then, this provides a simple way to do trade-offs. In light of this, the characteristics of Qpsq can be planned through the design of weighting functions, and they make it easy to consider the middle and high frequency part of Qpsq, which can guarantee the robustness, but the model error is amplified in some frequency bands, which restricts the closed-loop bandwidth. 4. System Design 4.1. Nominal Model One nominal model is needed in the presented hybrid control scheme, because both Qpsq and Kpsq are designed based upon it. From the analysis above, the difference between the nominal model and the actual plant, that is the uncertainty of the nominal model, determines the maximum allowable bandwidth, and therefore restricts the system performance. However, it can get a better result of design if the nominal model approximates the actual plant as far as possible. In our system, the derived equivalent linear models P0 psq with parameter variation are fifth-order models. The inverse of the fifth-order model will be extremely sensitive to high-frequency noise. Thus, a low-order model is appropriate; moreover, the low-order nominal model makes the design of the DOB simple. From Equation (9) and Figure 4, it is observed that system characteristics are mainly determined by the inertial frequency ωL of the hydraulic actuator, the second-order resonance frequency ωv of the servo valve, and the gain Kop . According to the idea of a dominant pole, if we take the middle value of ωL and Kop as nominal values and treat the second-order oscillation term of the hydraulic actuator as the perturbation, then a third-order nominal model Pn psq is obtained as Pn psq “

424908834 ps ` 38.82qps2 ` 1759s ` 1579000q

(24)

The uncertainty ∆L of the system, which is treated as a multiplicative perturbation can be calculated by ∆L psq “ Ppsq{Pn psq ´ 1 (25) 4.2. Design of Q(s) In the design of Qpsq, the weighting function W3 psq represents the upper limitation of the system uncertainty, and it can be designed according to the model uncertainty ∆L . In Figure 11,15 of 27  the black Appl. Sci. 2016, 6, 98  piq

lines represent the possible multiplicative perturbation ∆L of the models in P0 psq against the nominal

against  .  In possible order  to uncertainties cover  all  the  possible  ,  W s ) ,  the 11, is n (s)the L model Pn psq.the  Innominal  order tomodel  cover P all ∆L , W3uncertainties  psq, the blue line in3 (Figure blue line in Figure 11, is obtained as  obtained as (i)

piq

 10

  3 W 3 (“ s ) 1.914 1.914ˆ  10 ( s ps  1320) W3 psq 10´10 ` 1320q 3

(26) 

Disturbance Force Frequency Characteristic (i)

L

Magnitude (dB)

20

W3 0

-20

-40 0

10

1

10

2

10 Frequency (rad/s)

3

10

4

10

 

Figure 11. The defined system uncertainties and the  W3 (s) . 

Figure 11. The defined system uncertainties and the W3 psq.

The  weighting  W1 (s)   is  the  spectrum  characteristics  of  the  interference.  The  weighting  reciprocal  W11 ( s )   determines  the  ability  of  disturbance  suppression.  In  order  to  suppress  95%  of  the equivalent disturbance below 3 Hz,  W1 (s)   can be chosen as  W1 ( s ) 

0.64( s  262.5) 2   ( s  15) 2

(27) 

(26)

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The weighting W1 psq is the spectrum characteristics of the interference. The weighting reciprocal W1´1 psq determines the ability of disturbance suppression. In order to suppress 95% of the equivalent disturbance below 3 Hz, W1 psq can be chosen as W1 psq “

0.64ps ` 262.5q2

(27)

ps ` 15q2

By solving the mixed sensitivity problem described in Equation (23) with MATLAB Control System Toolbox (MathWork, New York, NY, USA, 2010) and Robust Control Toolbox (MathWork, New York, NY, USA, 2010), Kpsq is obtained. After being computed by Equation (21), Qpsq can be determined as Qpsq “

s5

` 485255s4

1.752 ˆ 1015 s ` 2.137 ˆ 1017 ` 2.293 ˆ 109 s3 ` 4.153 ˆ 1012 s2 ` 1.875 ˆ 1015 s ` 2.147 ˆ 1017

(28)

Figure 12a,b display the magnitude frequency response curves of Qpsq, W1´1 psq, 1 ´ Qpsq and condition and the interference suppression

W3´1 psq, respectively. Obviously, the robust stability Appl. Sci. 2016, 6, 98  requirement of the DOB system are both satisfied.

16 of 27  Bode Diagram 10

0

0

-10

-10

Magnitude (dB)

Magnitude (dB)

Bode Diagram 10

-20 -30 -40

-20 -30 -40

Q(s)

1-Q(s)

-1

-1

W3 (s) 1

10

2

W1 (s) 3

10 10 Frequency (rad/s)

4

10

1

10

2

3

4

10 10 Frequency (rad/s)

10

  ´1 W 1 ( s ) ;  (b)  Frequency  magnitude  (s)W Figure  (a) Frequency Frequency  magnitude  responses  of  Q   and  Figure12.  12. (a) magnitude responses of Qpsq and Frequency magnitude responses 3 3 psq; (b) ´1 1 of 1 ´ Qpsq and psq. W1 ( s ) .  1 QW(s1)   and  responses of 

As seen from the maximum frequency bandwidth of Qpsq is directly limited by the Meanwhile,  determines the ability of interference suppression. From Figure 12b, the  1 Figure Q ( s )   12a, weighting extraneous  function W3force  psq, and thus is very important determine oneby  appropriate 3 psq. that  is,  remaining  will  be itattenuated  to  −42 to dB  (decreased  98.7%)  at W least;  Meanwhile, 1 ´ Qpsq determines the ability of interference suppression. From Figure the combined  with  the  constant  velocity  compensator,  the  extraneous  force  will  be  eliminated 12b, almost  remaining extraneous force will be attenuated to ´42 dB (decreased by 98.7%) at least; that is, combined completely.  Other  low‐frequency  disturbance  will  also  be  suppressed  effectively.  Then,  the  with the constant velocity compensator, the extraneous force will be eliminated almost completely. disturbance will have little effect on the force‐tracking error.  Other low-frequency disturbance will also be suppressed effectively. Then, the disturbance will have little effect on the force-tracking error. 4.3. H∞ Controller  Q ( s )   is designed, the virtual plant of the outer loop is made certain, and then   1   can  4.3. After  H8 Controller be calculated, as is shown in Figure 13, in which  W 3 ( s )   is also drawn for comparison. In Figure 13,  After Qpsq is designed, the virtual plant of the outer loop is made certain, and then ∆1 can be the  black  possible    of  the  models  in  P0 ( s13, )  1(i) comparison. calculated,lines  as isrepresent  shown inthe  Figure 13, in multiplicative  which W3 psq isperturbation  also drawn for In Figure p i q against  the  nominal  model  is  compensated  by  the ∆DOB.  Compared  with  n (s) ,  which  L ,  the  the black lines represent the P possible multiplicative perturbation of the models in P psqagainst 1

0

uncertainty  restrained  significantly  in  the  low‐frequency  region,  while  and  the nominalis  model Pn psq, which is compensated by the DOB. Compared with in  ∆L ,the  the middle‐  uncertainty is high‐frequency regions, the uncertainty enlarges some, but it is still below  restrained significantly in the low-frequency region, while in the middle- andWhigh-frequency regions, 3 ( s ) . This means the  the uncertainty enlarges some, but itchanges  is still below W3 psq.being  This means the allowable ofthe  the allowable  bandwidth  of  the  system  little after  compensated  by  the bandwidth DOB,  while  system  perturbation  decreases  greatly  in  the  low‐frequency  region.  The  function  W 3 ( s )   can  be  a  candidate weighting function to describe the uncertainty of the outer loop.  System Uncertainties after Compensation

After  Q ( s )   is designed, the virtual plant of the outer loop is made certain, and then   1   can  be calculated, as is shown in Figure 13, in which  W 3 ( s )   is also drawn for comparison. In Figure 13,  the  black  lines  represent  the  possible  multiplicative  perturbation  1(i)   of  the  models  in  P0 ( s)   against  the  nominal  model  Pn (s) ,  which  is  compensated  by  the  DOB.  Compared  with   L ,  the  Appl. Sci. 2016, 6, 98 of 26 uncertainty  is  restrained  significantly  in  the  low‐frequency  region,  while  in  the  middle‐  16 and 

high‐frequency regions, the uncertainty enlarges some, but it is still below  W 3 ( s ) . This means the  allowable  bandwidth  of  the  system  changes  little after  being  compensated  by  the  DOB,  while  the  system changes little after being compensated by the DOB, while the system perturbation decreases system  perturbation  decreases  greatly  in  the  low‐frequency  region.  The  function  W 3 ( s )   can  be  a  greatly in the low-frequency region. The function W3 psq can be a candidate weighting function to candidate weighting function to describe the uncertainty of the outer loop.  describe the uncertainty of the outer loop. System Uncertainties after Compensation 20

Magnitude (dB)

0 -20 -40 (i)

-60

1

-80

W3

0

10

1

10

2

10 Frequency (rad/s)

3

10

4

10

 

Figure 13. System uncertainties for outer loop design.  Figure 13. System uncertainties for outer loop design. Appl. Sci. 2016, 6, 98 

17 of 27 

The H8 mixed sensitivity method is used to derive the performance controller Gc psq. The  H∞  mixed  sensitivity  is  used  to  derive  the  performance  The  are G c ( s ) . loop The sensitivity function Sc andmethod  the complementary sensitivity function controller  TC of the outer sensitivity  function  the  complementary  sensitivity  deduced according to Sthe definition of robust control theory asfunction  T C   of  the  outer  loop  are  c   and  deduced according to the definition of robust control theory as  Pn psqGc psq 1 , TC “ Pn ( s )Gc ( s ) (29) SC “ 1 1 ` P psqG psq 1 ` Pn psqG c psq SC  n c , TC  (29)  1  Pn ( s )Gc ( s )

1  Pn ( s )Gc ( s )

The performance weighting function W1o psq of the outer loop is set as The performance weighting function  W1o ( s )   of the outer loop is set as  3.017 s ` 17500 3.017 s + 17500 W W1o1o(psq s ) “ 33.33 s ` 1   33.33 s + 1

(30) 

(30)

Set 3 psq, W2o psq “ 0.00001, and then solving the standard H8 mixed sensitivity Set WW3o3opsq (s) “  WW 3 ( s ) ,  W 2o ( s )  0.00001 ,  and  then  solving  the  standard  H∞  mixed  sensitivity  problem, G psq is obtained as problem,  cG c ( s )   is obtained as  ) GcGpsq c ( s“

2 4809662.1287ps `38.82)( 38.82qps 1759s ` 1579000q 4809662.1287( s  s2 ` 1759 s  1579000)   2 2  426300)( s `0.03)( s  4509 s  739900 0) ps( s` 426300qps 0.03qps ` 4509s ` 7399000q

(31)  (31)

The closed‐loop characteristic curve of the nominal plant is displayed in Figure 14.  The closed-loop characteristic curve of the nominal plant is displayed in Figure 14.

Magnitude (dB)

0

-20

-40

Phase (deg)

-60 0

-90

-180

-270 0 10

 

1

10

2

10 Frequency (rad/s)

3

10

4

10

 

Figure 14. Closed‐loop characteristic curve of the nominal plant.  Figure 14. Closed-loop characteristic curve of the nominal plant.

5. Simulation Experiment  In  the  above sections,  the  outer  loop  controller  and  the  disturbance  compensator  have  been  designed  based on the established linear mathematical model with  perturbations.  The influence  of  some factors has been neglected, and part of the system nonlinearity has been treated by equivalent 

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5. Simulation Experiment In the above sections, the outer loop controller and the disturbance compensator have been designed based on the established linear mathematical model with perturbations. The influence of some factors has been neglected, and part of the system nonlinearity has been treated by equivalent linearization. In order to validate the effectiveness of the designed hybrid controller, a simulation model that is close to the practical situation is needed. So, in the simulation experiment section, the model of the loading system is built in AMESim (LMS Imagine, Leuven, Belgium, 2010), as depicted in Figure 15a; it contains the nonlinear characteristics of the load flowrate of the servo valve and the un-modeled dynamics, and it is closer to the actual plant than the linear model. The developed algorithm is conducted in MATLAB/Simulink (MathWork, New York, NY, USA, 2010). Through co-simulation using AMESim and MATLAB/Simulink as shown in Figure 15b, the performance of the designed hybrid controller can be evaluated. The schematic diagrams of co-simulation are shown in Figure 15. Appl. Sci. 2016, 6, 98  18 of 27 

xM s

Fr

Gc ( s)

FL

Gcom (0)

Ka

Q( s )

iv

Q( s )/Pn ( s )

  Figure 15. Schematic diagrams of co‐simulation: (a) AMESim block diagram; and (b) co‐simulation  Figure 15. Schematic diagrams of co-simulation: (a) AMESim block diagram; and (b) co-simulation in in MATLAB/Simulink.  MATLAB/Simulink.

In Figure Figure 15a, 15a,  the  gain  block  with  a  value  of  1000  is  used  to  convert  the  of  the current control  In the gain block with a value of 1000 is used to convert the unit of unit  the control current of the servo valve from amperes (A) in MATLAB to milliamperes (mA) in AMESim, and the  of the servo valve from amperes (A) in MATLAB to milliamperes (mA) in AMESim, and the gain gain for the signal output of the force sensor is set to −1 to make the output direction of the force  for the signal output of the force sensor is set to ´1 to make the output direction of the force sensor sensor consistent with Equation (1).  consistent with Equation (1). In the the AMESim AMESim model, model, besides besides the the nonlinear nonlinear dynamic dynamic characteristics characteristics of of the the servo servo valve, valve, the the  In following factors are considered: the supply pressure; the internal leakage, static friction, Coulomb  following factors are considered: the supply pressure; the internal leakage, static friction, Coulomb friction and viscous friction of the loading cylinder; the equivalent mass of the load; and the oil bulk  friction and viscous friction of the loading cylinder; the equivalent mass of the load; and the oil bulk modulus. Parameters related to the loading cylinder are chosen according to the characteristics of  modulus. Parameters related to the loading cylinder are chosen according to the characteristics of the servo hydraulic cylinder in the industrial environment. A friction model with the Stribeck effect  the servo hydraulic cylinder in the industrial environment. A friction model with the Stribeck effect is designed. The used parameters in AMESim are listed in Table 2 and are consistent with those in  is designed. The used parameters in AMESim are listed in Table 2 and are consistent with those in Table 1.  Table 1. Table 2. Parameters in AMESim.  Table 2. Parameters in AMESim. Parameter  Parameter Supply pressure  p s   Supply pressure p Maximum flow rate  s Maximum flow rate Rated pressure drop  Rated pressure drop Valve rated current  Valve rated current Valve natural frequency  Valve natural frequency Valve damping ratio  Valve damping ratio Piston diameter  Piston diameter Rod diameter Rod diameter  Full stroke Full stroke 

Value Value 180–210 bar  180–210 bar 40 L/min  40 L/min 70 bar  70 bar 20 mA  20 mA 200 Hz  200 Hz 0.7 0.7 70 mm  70 mm 42 mm 42 mm  150 mm 150 mm 

Parameter Parameter Clearance on diameter  Clearance on diameter Coefficient of viscous friction  Coefficient of viscous friction Maximum static force  Maximum static force Maximum Coulomb friction  Maximum Coulomb friction Force sensor stiffness  Force sensor stiffness Dead volume  Dead volume Mass  MassmmLL  Oil modulusβe e  Oil modulus  Oil density Oil density 

Value  Value 0.02 mm  0.02 mm 800 N/(m/s)  800 N/(m/s) 150 N  150 N 100100 N  N 8 N/m  8 N/m 1 × 10 1 ˆ 10 3 40 cm3  40 cm 8–12 kg  8–12 kg 7000–11,000 bar 7000–11,000 bar  870 kg/m3 3 870 kg/m  

In  the  course  of  co‐simulation,  all  experiments  are  performed  under  the  typical  operating  condition, i.e., in the presence of motion disturbance, and some unified settings are made: (1) The  motion  disturbance  input  is  set  as  xM   =  7sin14t  (mm/s)  unless  otherwise  stated  to  produce  the  largest motion disturbance. Its magnitude is higher than the maximum magnitude when the motion 

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In the course of co-simulation, all experiments are performed under the typical operating condition, i.e., in the presence of motion disturbance, and some unified settings are made: (1) The motion disturbance input is set as xM = 7sin14t (mm/s) unless otherwise stated to produce the largest motion disturbance. Its magnitude is higher than the maximum magnitude when the motion . disturbance is at its maximum frequency. (2) The xM is acquired by the direct differential method. (3) The nominal values of the perturbed parameters ps , βe , and mL are 21, 900 MPa, and 10 kg, respectively. (4) The AMESim print interval is set as 0.0001 s. 5.1. Disturbance Suppression without Loading Force Command As noted above, the ability of disturbance suppression of the loading system has a severe effect on the force-tracking performance. To verify the ability of disturbance suppression, the loading force command Fr is always set to zero in this section. 5.1.1. Suppression Performance In the proposed hybrid control scheme, the extraneous force is gradually eliminated by the composite methods. Therefore, four kinds of experiments are successively made to verify their corresponding suppressing effects. All experiments are conducted under the conditions that the motion disturbance input is 7sin14t (mm/s), the force feedback loop is always closed and all perturbed parameters take their nominal values. (1)

(2) (3) (4)

The original extraneous force is tested for comparison in the condition that the outer loop controller Gc psq “ 1 and without the constant compensation term and DOB inner loop compensator. The suppression by using the constant compensator. In this case, there is no DOB inner loop compensator and the outer loop controller Gc psq “ 1. The suppression by using the constant compensator combined with the DOB inner loop compensator. In this case, the outer loop controller Gc psq “ 1. The suppression by using the designed hybrid controller.

The experimental results are drawn in the form of a Lissajous curve, and are depicted in Figures 16 and 17 because xM , xL and FL have the same frequency when the loading force command equals zero. In Figure 16, the curves of FL and xL to xM have a clear physical meaning, and they illustrate the formation of the output force which is defined by Equation (1). In this case, the output force is the extraneous force. The smaller the spacing in the middle region of the curve of xL to xM , the smaller . the extraneous force is. In Figure 17, the curve of FL to xM illustrates the relationship between the disturbance velocity and the extraneous force, and provides the information about when and how the extraneous force gets its maximum value. This curve is useful for evaluating the suppressing effect on the extraneous force. As seen from Figures 16 and 17 the original extraneous force is very big (up to 13,310 N, in Figures 16a and 17a). It is far beyond other disturbances in the loading system, and even more than some command loading forces. It is meaningful to suppress the extraneous force. After being compensated by the constant compensator, the extraneous force is decreased to 2688 N (decreased by 80%, in Figures 16b and 17a). This result is very close to the analysis result about the suppressing ability of the constant compensator in Section 3.1 (decreased by 84%). By using the constant compensator combined with the DOB inner loop compensator, the extraneous force is further reduced to 102 N (decreased by 96.2% from 2688 N, in Figures 16c and 17b). This result is close to our design in Section 4.2 where the designed Q-filter can attenuate the disturbance up to ´42 dB (decreased by 98.7%). With the designed hybrid controller, the extraneous force is restrained to 57 N (decreased by 99.6% from the original extraneous force, in Figures 16d and 17b). This implies that the proposed hybrid control scheme and controller are valid, and achieve excellent suppression of the disturbance force (less than 60 N) even with the largest motion disturbance.

Appl. Sci. 2016, 6, 98 Appl. Sci. 2016, 6, 98  Appl. Sci. 2016, 6, 98 

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   x –– xxM    with  Figure  16.  Relationship  diagrams  of  7sin14t  (a)  LL – Figure  Relationship diagrams diagrams of of FLFF – xxMM    and  and  with disturbance  disturbance  7sin14t (mm/s),  (mm/s),  (a)  Figure 16.  16. Relationship –x xL –xMxLLwith 7sin14t (mm/s), (a) original M disturbance M and original  extraneous  force  (b)  by  compensator  (c)  by  extraneous force case; (b)case;  restrained by constant compensator case; (c)case;  restrained by constant original  extraneous  force  case;  (b)  restrained  restrained  by  constant  constant  compensator  case;  (c)  restrained  restrained  by  constant compensator with DOB case; (d) restrained by hybrid controller case.  compensator with DOB case; (d) restrained by hybrid controller case. constant compensator with DOB case; (d) restrained by hybrid controller case.  by by constant constant compensator compensator and and DOB DOB by by hybrid hybrid controller controller

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F . – xxM    with  Figure  17.  Relationship  diagrams  disturbance  7sin14t  (mm/s),  (a)  Figure  17. Relationship Relationship diagrams diagrams ofof  of  disturbance  7sin14t  (mm/s),  (a)  comparative  comparative  Figure 17. FL –FxLLM– with disturbance 7sin14t (mm/s), (a) comparative results M with  results under different cases; (b) enlarged graph of results of constant compensator with DOB case  under different cases; (b) enlarged graph of results of constant compensator with DOB case and hybrid results under different cases; (b) enlarged graph of results of constant compensator with DOB case  and hybrid controller case.  controller case. and hybrid controller case. 

Notice that in Figures 16c, d and 17b, there are obvious burrs and peak values on the curves of  Notice that in Figures 16c, d and 17b, there are obvious burrs and peak values on the curves of  Notice that in Figures 16c,d and 17b, there are obvious burrs and peak values on the curves .   to     to  . .  The  high  burrs  on  the     to    appear  at  Fof   Lto toxxMxMM   and  and  to  The  high  burrs  on curve the  curve  curve  of xFM to  xxMM nearly appear  nearly  at  the  the  FLL F FLL appear and FFFLLL to xMxx.MMThe high burrs on the of FL of  to at nearly  the maximum . .   maximum  displacement,  and  the  sharp  peaks  on  the  curve  of    to    occur  around  F x x  0 .  displacement, and the sharp peaks on the curve of F to x occur around x “ 0. This implies there   maximum  displacement,  and  the  sharp  peaks  on  Lthe  curve  of  FLL   to  xMM M M   occur  around  xM M  0 .  This  This  implies  implies  there  there  is  is  a  a  relationship  relationship  between  between  the  the  generation  generation  of  of  sharp  sharp  force  force  disturbance  disturbance  and  and  the  the 

Appl. Sci. 2016, 6, 98 Appl. Sci. 2016, 6, 98 

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is a in  relationship between the generation of characteristic  sharp force disturbance theversus  changes in the direction changes  the  direction  of  motion.  From  the  of  friction and force  velocity  x L   of  of . motion. From the characteristic of friction force versus velocity x of the hydraulic cylinder, as L is  changed  sharply  when  depicted the  hydraulic  cylinder,  as  depicted  in  Figure  18a,  the  friction  force  the  . in Figurex 18a, the friction force is changed sharply when the direction of xL changes. Moreover, direction of    changes. Moreover, from the corresponding graph of the friction force and  FL   in from L the corresponding graph of the friction force and F in the time domain, as depicted in Figure L the  time  domain,  as  depicted  in  Figure  18b,  the  start  time  of  the  peak  of  the  force  output  is  just  18b, the start time of the peak of the force output is just corresponding to the time when the friction force corresponding to the time when the friction force is changing. Thus, this sharp force disturbance is  is changing. this sharp forceforce  disturbance is mainlycylinder  caused by theis impact friction force of the mainly  caused  by Thus, the  impact  friction  of  the  hydraulic  that  generated  when  the  hydraulic cylinder that is generated when the motion is reversing. motion is reversing. 

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Figure 18. Effect of friction on suppression, (a) friction model with Stribeck effect; (b) corresponding  Figure 18. Effect of friction on suppression, (a) friction model with Stribeck effect; (b) corresponding graph of friction force and  FL   in the time domain with disturbance 7sin14t (mm/s).  graph of friction force and FL in the time domain with disturbance 7sin14t (mm/s). . It  can  be  verified  that  the  curves  of  FL   to  xM   and  FL   to  x M   are  close  to  ellipses  in  the  It can be verified that the curves of FL to xM and FL to xM are close to ellipses in the absence of absence  of  friction  force.  Without  the ofimpact  friction  force,  the  output maximum  will  be    16c friction force. Without the impact frictionof  force, the maximum forceoutput  will beforce  20 N in Figure 20 N in Figure 16c and 5 N in Figure 16d, and that means the extraneous force resulting from the  and 5 N in Figure 16d, and that means the extraneous force resulting from the disturbance motion disturbance motion  is successfully  suppressed.  Simultaneously,  is  known  Figures  and  is successfully suppressed. Simultaneously, it is known fromit Figures 17bfrom  and 18b that 17b  the designed 18b  that  the  designed  hybrid  controller  has  a  certain  restraining  ability  on  friction  force  (reduced  hybrid controller has a certain restraining ability on friction force (reduced from 150 to 57 N), but the from 150 to 57 N), but the restraining effect is worse than that of the extraneous force. In order to  restraining effect is worse than that of the extraneous force. In order to verify our design of the hybrid verify  our  design  the  hybrid  controller,  the  assumed  force  is  larger  the  actual  controller, the of  assumed friction force is larger than the friction  actual low-friction servothan  hydraulic cylinders. low‐friction servo hydraulic cylinders. Therefore, in practical application, the impact of the friction  Therefore, in practical application, the impact of the friction force will be relatively small. force will be relatively small.  5.1.2. Robustness of Suppression 5.1.2. Robustness of Suppression  When designing the controllers, the perturbations of some system parameters have been taken into When designing the controllers, the perturbations of some system parameters have been taken  consideration. The designed hybrid controller achieves a good disturbance suppression performance into  with consideration.  The  designed  hybrid  controller  a  good  disturbance  suppression  the nominal values of perturbation parameters.achieves  The influence of each perturbation parameter on performance  with  the  nominal isvalues  perturbation  parameters.  influence  of  each  the the disturbance suppression studiedof  separately, meaning that whenThe  a parameter is perturbed, perturbation parameter on the disturbance suppression is studied separately, meaning that when a  other parameters are taking their nominal values. The experiments are carried out with the designed parameter is perturbed, the other parameters are taking their nominal values. The experiments are  hybrid controller. carried out with the designed hybrid controller.  Figure 19a–c show the restraining results for different ps , βe and mL , respectively. It is observed Figure  show  the  results  for  different    and  It  output is  p s ,   mL ,  respectively.  that the19a–c  perturbation of prestraining  a very small increase ine the maximum value of the s and βe causes force (less than 5 N), while there almost effect for the perturbation of mL . It can also be known observed that the perturbation of  causes a very small increase in the maximum value  p s  isand   e   no from Figure 19b that increasing fluid bulk modulus βe is beneficial to disturbance suppression. of the output force (less than 5 N), while there is almost no effect for the perturbation of  mL . It can  Therefore, we can draw that the disturbance force, including the extraneous force and the friction also be known from Figure 19b that increasing fluid bulk modulus   e   is beneficial to disturbance  force, is effectively suppressed by the proposed hybrid control scheme and controller, and it has suppression.  considerable robustness within the perturbation range of our study. Therefore,  we  can  draw  that  the  disturbance  force,  including  the  extraneous  force  and  the  friction force, is effectively suppressed by the proposed hybrid control scheme and controller, and it  has considerable robustness within the perturbation range of our study. 

Appl. Sci. 2016, 6, 98 Appl. Sci. 2016, 6, 98 

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5.2. Loading Force Tracking  5.2. Loading Force Tracking 5.2.1. Typical Loading Force Case 5.2.1. Typical Loading Force Case  Thecommand  command loading  loading forces verify the tracking performance of the designed H8 The  forces are are applied applied toto  verify  the  tracking  performance  of  the  designed  feedback controller, because the strong disturbance force has already been effectively restrained by H∞ feedback controller, because the strong disturbance force has already been effectively restrained  the compensators. by the compensators.  Two typical types of loading force spectrum are acted on the system. They are all in the form of Two typical types of loading force spectrum are acted on the system. They are all in the form of  dynamic load superimposed on a large static load, and they are different in amount and frequency of dynamic load superimposed on a large static load, and they are different in amount and frequency  dynamic load. OneOne  is 10is  ` 5sin80πt (kN), and the other 4 ` sin160πt (kN). During theDuring  experiment, 10  5sin80 t   (kN),  t   (kN).  and  the isother  is  4  sin160 the  of  dynamic  load.  the motion disturbance is always exerted on the loading system. At the same time, the influence experiment, the motion disturbance is always exerted on the loading system. At the same time, the  on force tracking by the perturbation parameters is also taken into account. For different command influence on force tracking by the perturbation parameters is also taken into account. For different  loading forces, the curves of the  FL and xL to xof  regular because theirnot  frequencies not the their  same. command  loading  forces,  curves    and  regular are because  FL not x L   to  xM   are  M are Therefore, we analyze the loading performance in the time domain. The results of two kinds of loading frequencies are not the same. Therefore, we analyze the loading performance in the time domain.  forces are shown in Figures 20 and 21 respectively. The results of two kinds of loading forces are shown in Figures 20 and 21, respectively.  From the experimental results, there are some relatively consistent results. The perturbation of mL has almost no effect on the load output. The perturbation of ps has a small effect on the load output, and the best tracking performance is achieved at the maximum supply pressure. The perturbation of βe has an obvious effect on the load output, and when βe equals 700 MPa the maximum magnitude error to the loading force command reaches 5.2% in Figure 20a and 3% in Figure 21a, respectively. This is because larger ps and βe increases the open loop bandwidth of the system, and thus the error between the physical plant and the nominal model will be reduced, and that means the H8 controller can use more gain in tracking. It can be verified that if the βe is set to 1100 MPa, the output error will be less than 2% for all the above experiments, and this can be easily implemented in engineering. So, the tracking performance of the designed H8 controller can be guaranteed in the range of allowable perturbations.

Appl. Sci. 2016, 6, 98 Appl. Sci. 2016, 6, 98 

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5.2.2.From the experimental results, there are some relatively consistent results. The perturbation of  Small Command Loading Force Case mL   has almost no effect on the load output. The perturbation of  p s   has a small effect on the load  Our typical loading force spectrums are relatively large, and are much larger than the friction output,  and  the ofbest  performance  is  achieved  at easily the  maximum  supply  force. The effect the tracking  friction force on the output force is not observed. Then, wepressure.  performedThe  the perturbation  of    has  an  obvious  effect  on  the  load  output,  and  when    equals  700  MPa force the    e a slightly experiments withea small command loading force of 200 ` 100sin10πt (N) and larger maximum  magnitude  error  to  the  loading  force  command  5.2% allin perturbation Figure  20a  and  3%  in  with the same frequency of 500 ` 500sin10πt (N). During the reaches  experiment, parameters Figure 21a, respectively. This is because larger    and    increases the open loop bandwidth of  ps  e exerts on them. take their nominal values and the motion disturbance always

the system, and thus the error between the physical plant and the nominal model will be reduced,  and that means the H∞ controller can use more gain in tracking. It can be verified that if the   e   is  set to 1100 MPa, the output error will be less than 2% for all the above experiments, and this can be 

5.2.2. Small Command Loading Force Case  Our typical loading force spectrums are relatively large, and are much larger than the friction  force. The effect of the friction force on the output force is not easily observed. Then, we performed  the experiments with a small command loading force of  200 100sin10t   (N) and a slightly larger  Appl. Sci. 2016, 6, 98 23 of 26 force  with  the  same  frequency  of  500  500sin10t   (N).  During  the  experiment,  all  perturbation  parameters take their nominal values and the motion disturbance always exerts on them.  From the results in Figure 22a,b, friction has a certain effect on the output of a small command  From the results in Figure 22a,b, friction has a certain effect on the output of a small command loading force, but the output shape is still maintained. In addition, that effect is not obvious for a  loading force, but the output shape is still maintained. In addition, that effect is not obvious for a slightly larger command loading force.  slightly larger command loading force. Force output

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  Figure  22. The The  loading  performance  small  command  loading  force,  of (a)  of  Figure 22. loading performance of smallof command loading force, (a) command 200command  ` 100sin10πt 200 (b) 100sin10 t   (N); (b) command of  500  500sin10t   (N).  (N); command of 500 ` 500sin10πt (N).

5.2.3. Comparative Results of Different Outer Loop Controllers  5.2.3. Comparative Results of Different Outer Loop Controllers There are two purposes for the design of the DOB inner loop compensator. One is to suppress  There are two purposes for the design of the DOB inner loop compensator. One is to suppress the disturbance including the extraneous force and friction. The other is to compensate the model  the disturbance including the extraneous force and friction. The other is to compensate the model mismatch  mismatch between  between the  the nominal  nominal model and  model and the  the controlled  controlled plant, and  plant, and to  to make  make the  the controlled  controlled plant  plant behave like the nominal model in a certain frequency range. Through the above analysis, the DOB  behave like the nominal model in a certain frequency range. Through the above analysis, the DOB compensated  plant can can bebe  treated  perturbation  but  no  disturbance.  Therefore,  we  Pn (with s)   with  compensated plant treated as as  Pn psq perturbation but no disturbance. Therefore, we design design  outer  PID  to controller  to  compare  with  the ofperformance  designed  H∞  an outeran  loop PID loop  controller compare with the performance the designed of  H8the  controller. controller.  The parameters of the PID controller for this experiment are proportional gain 2.48, integral of  the gain PID 0.controller  this  experiment  are  proportional gain  gain The  55.78parameters  and differential This PIDfor  controller and the designed H8 controller2.48,  haveintegral  almost gain 55.78 and differential gain 0. This PID controller and the designed H∞ controller have almost  the same rise time of the step response of the nominal model Pn psq. Then, we measure the output the same rise time of the step response of the nominal model  . Then, we measure the output of  Pn (s)by of the command loading force on the controlled plant modeled AMESim and Pn psq separately, compare the output of the same under plant  a different model/plant, and compare output of   separately,  the  command  loading  force  on controller the  controlled  modeled  by  AMESim  and  Pn (s)the the different controllers under the same model. The command loading forces are 1 ` 4sinpi ˆ 2πtq (kN), compare the output of the same controller under a different model/plant, and compare the output  where i “ 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. In the AMESim model, the perturbation parameters of the different controllers under the same model. The command loading forces are  1  4 sin(i  2take t )   their nominal value and the motion disturbance always exerts on it. The maximum magnitudes of the (kN),  where  i  5,10,20,30,40,50,60,70,80,90,100 .  In  the  AMESim  model,  the  perturbation  output results are listed in Table 3. parameters take their nominal value and the motion disturbance always exerts on it. The maximum  magnitudes of the output results are listed in Table 3.  Table 3. Comparative results of different outer loop controllers. As seen from the data in Table 3, the performance of the PID controller for the controlled plant  is good below 50 Hz. The outputs of  Pn (s)   and the controlled plant modeled by AMESim are close,  Outer loop PID Controller

H8 Controller

controller and they are very similar with the results of the H∞ controller. That is because the model mismatch  and the disturbance below 50 Hz have been compensated by the DOB compensator, and this result  Output of Output of Relative Output of Output of Relative Frequency Pn psqdesign  (N) AMESim (N) The  Erroroutput  (N) Pn psq between  (N) AMESim (N) the Error (N) is  consistent  with  our  of  the  Q‐filter.  error  controlled  Pn (s)   and  5 Hz PID  increases  4984 with  a  frequency  4992 4999 5005 6 has  a  plant  with  above 850  Hz,  and  the  magnitude  of  the  output  10 Hz 4980 5004 24 4998 5010 12 trend of divergence. For the H∞ controller, the output error between  Pn (s)   and the controlled plant  20 Hz

4983

5005

22

4996

5013

17

40 Hz 50 Hz 60 Hz 70 Hz 80 Hz 90 Hz 100 Hz

5010 5031 5058 5082 5120 5157 5237

5100 5171 5255 5352 5450 5535 5600

90 140 197 270 330 378 363

4984 4975 4964 4951 4936 4919 4900

5075 5100 5130 5137 5120 5090 5050

91 125 166 186 184 171 150

increases a little but starts to decrease about 70 Hz; meanwhile, the magnitude of the output has a  30 Hz 4994 5043 49 4991 5037 46

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As seen from the data in Table 3, the performance of the PID controller for the controlled plant is good below 50 Hz. The outputs of Pn psq and the controlled plant modeled by AMESim are close, and they are very similar with the results of the H8 controller. That is because the model mismatch and the disturbance below 50 Hz have been compensated by the DOB compensator, and this result is consistent with our design of the Q-filter. The output error between Pn psq and the controlled plant with PID increases with a frequency above 50 Hz, and the magnitude of the output has a trend of divergence. For the H8 controller, the output error between Pn psq and the controlled plant increases a little but starts to decrease about 70 Hz; meanwhile, the magnitude of the output has a trend of convergence. That means the designed H8 controller indeed guarantees system robustness in high-frequency regions while ensuring the tracking performance. For the control effects under the parameter perturbations, they can be verified by similar processing as used in the above method. At present, the effectiveness of the designed hybrid controller has been verified by using numerical simulations. The future work is to conduct the related computer control for the real loading system. 6. Conclusions Through the analysis, discussion and co-simulation based on AMESim and MATLAB/Simulink, the following conclusions can be summarized as: (1)

(2)

(3)

The handling method of the linear models set can describe the nonlinearity, parameter uncertainties and strong interference characteristics of the hydraulic servo loading system, and it can provide an available method to determine the weighting function to express the uncertainty. The proposed design method can design a hybrid, robust controller that consists of a constant velocity feed-forward compensator, a disturbance observer–based compensator and a robust H8 feedback controller. This hybrid, robust controller can eliminate the motion disturbance and compensate the model perturbation caused by nonlinearity and parameter variation in the electro-hydraulic system. The proposed solution to determine the low-pass filter Qpsq in the disturbance observer based on H8 mixed sensitivity theory is valid.

In future work, the computer control realization of the proposed robust control law for the real loading system and the nonlinear control law will be conducted. Acknowledgments: The authors would like to acknowledge the support of National Natural Science Foundation of China (Grant No. 51475019) and National Key Basic Research Program of China (Grant No. 2014CB046403). Author Contributions: Design of the research scheme and research supervision: Yunhua Li; Performing of the research and writing of the manuscript: Zhingqing Sheng. Conflicts of Interest: The authors declare no conflict of interest.

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