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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Appl. Sci. 2016, 6, 98 points. When the
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 4e 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 4L 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 4e K tmLK s L s3 K`s 4Le K K tm K sL `4e 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 andvζv (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 .
xM
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 4e AL2 /VL is the liquid spring stiffness, 0 4e 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 W11 ( 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 psqagainst 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
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Figure 13. System uncertainties for outer loop design. Figure 13. System uncertainties for outer loop design. Appl. Sci. 2016, 6, 98
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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.
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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
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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 modulusβe 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.
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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
Original Originalextraneous extraneousforce force by byconstant constantcompensator compensator
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F . – xxM 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 xMxx.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
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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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Figure 19. The restraining effects on disturbance, (a) with perturbation of ; (b) with perturbation Figure 19. The restraining effects on disturbance, (a) with perturbation of pps ;s (b) with perturbation of of mLm ; (c) with perturbation of βe . e . L ; (c) with perturbation of
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.
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5sin80tpkNq, (kN) , (a) Figure loading performance performance of of loading loading force force (a) with perturbation Figure20. 20. The The loading 1010 `5sin80πt with perturbation of of ps ; p m ; (b) with perturbation of ; (c) with perturbation of . s L e s L e (b) with perturbation of m ; (c) with perturbation of β . L
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sin160tpkNq, (kN) , (a)(a) Figure of loading loading force force 4 4 with perturbation Figure21. 21. The The loading loading performance performance of `sin160πt with perturbation of of ps ; p ss ; (b) with perturbation of mLLwith ; (c) with perturbation of (b) with perturbation of mL ; (c) perturbation of βe . ee .
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 100sin10t (N) and a slightly larger Appl. Sci. 2016, 6, 98 23 of 26 force with the same frequency of 500 500sin10t (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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4
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 500sin10t (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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