- Email: [email protected]
Compound Control for Hydraulic Flight Motion Simulator
Chinese
Journal of Aeronautics
Chinese Journal of Aeronautics 23(2010) 240-245
www.elsevier.com/locate/cja
Compound Control for Hydraulic Flight Motion Simulator Wang Benyonga,*, Dong Yanliangb, Zhao Kedingb a
School of Mechanical Engineering, Heilongjiang Institute of Science and Technology, Harbin 150027, China b
School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China Received 20 February 2009; accepted 30 April 2009
Abstract The design of a compound control is presented for the servo system of hydraulic flight motion simulator, which suffers from highly nonlinear dynamics, large parameter time-variation and severe load coupling among channels. The compound control is composed of a robust feedback controller and a feedforward compensator. The design aim is to achieve high tracking performance even in the presence of considerable uncertainty, external disturbance and load coupling among channels. Toward this aim the feedback controller for rejecting perturbation and disturbance is designed by using μ synthesis optimization technique and the feedforward compensator for compensating time lag of dynamic system is established based on the basic idea of zero phase error tracking. To validate the proposed control strategy, simulations and experiments are implemented, and show that the resulting system is highly robust against model perturbation and possesses excellent capability of suppressing the load coupling and improving the tracking performance. Keywords: hydraulic flight motion simulator; system identification; compound control; μ synthesis; feedforward compensation
1. Introduction1 Hydraulic flight motion simulator (HFMS) is a three-degree-of-freedom complex high-accuracy mechanism as shown in Fig.1. HFMS has been widely used in flight motion simulation of various aircraft because it possesses greater power to weight ratio, higher stiffness and less motion error than its electrical rival[1].
1, 8, 11, 15—Servo motor; 2, 7, 10, 14—Angular sensor; 3, 6, 9, 13—Electro-hydraulic servo valve; 4—Middle gimbal; 5—Inner gimbal; 12—Outer gimbal; 16—Base
Fig.1
Schematic structure of a HFMS.
*Corresponding author. Tel.: +86-451-88036252. E-mail address: [email protected] Foundation item: National “211 Project” Foundation of China 1000-9361/$ - see front matter © 2010 Elsevier Ltd. All rights reserved. doi: 10.1016/S1000-9361(09)60211-9
Although HFMS has many distinct advantages, the characteristics of highly nonlinear dynamics, large parameter time-variation and severe load coupling disturbance among channels[2] make various controller design techniques, such as proportion integral derivative (PID), lead-lag compensation, adaptive control[3], sliding model control[4] and others, not be suitable for such high-accuracy plant. The μ synthesis optimization technique[5] can excellently solve this problem due to its extremely effective robustness and enables the threechannel controllers of HFMS to be designed separately. Considering the characteristics of HFMS and the requirement for high tracking performance even in the presence of considerable uncertainty, external disturbance and load coupling among channels, a compound control for HFMS electro-hydraulic servo system is presented in this article. The compound control consists of a robust feedback controller and a feedforward compensator. The feedback controller for attenuating perturbation and disturbance is designed by using μ synthesis optimization technique. The feedforward compensator for compensating time lag of dynamic system is established based on the basic idea of zero phase error tracking. The simulation and experiment results show that the proposed compound control for HFMS servo system is highly robust against large perturbation and extremely effective for suppressing the
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severe load-coupling disturbance among channels and improving system trajectory tracking performance. 2. Modeling The single channel system model for HFMS can be written as[6]
G = G0 + Wm Δg =
Kv
+ Wm Δg (1) §s · 2ξ h s¨ 2 + s + 1¸ © ωh ωh ¹ where G is the real system model, G0 the nominal model, Ȧh the hydraulic undamped natural frequency, ȟh the hydraulic damping ratio, Kv the open-loop gain, s the Laplace operator, Wm Δg the additive error of system model, Wm the weighting function reflecting the amount of uncertainty in the model and Δg denotes the normalized uncertainty satisfying ||Δg||<1. The modeling is accomplished by using system identification[7]. To collect the sufficiently informative data for model identification, the persistently exciting input is chosen to be a pseudo-random binary signal (PRBS) which has the clock time of 0.009 s, the period length of 2 046 and shift between −0.5 V and 0.5 V. A part of the PRBS is shown in Fig.2. The identification procedure is illustrated with the HFMS inner gimbal servo system. 2
Fig.3
Output signals of inner gimbal single channel system.
Fig.4
Examination of inner gimbal system model.
2.2. Estimation of model uncertainty Here the additive error Wm Δg, namely, model uncertainty, is modeled by model error modeling technique based on prediction error algorithm[8]. The dynamic model of Wm Δg can be expressed by A( y − G0u ) =
Fig.2
A part of PRBS for system identification.
2.1. Parameter identification of nominal model As the first step, the above PRBS is input to the inner gimbal single channel system of HFMS, and the output data are recorded and preprocessed as shown in Fig.3. Then the preprocessed data are split into two halves, the first to be used for estimation, and the second one for validation. Based on the estimation data the best nominal model can be computed with prediction error identification algorithm. The obtained identification parameters for inner gimbal servo system are Ȧh = 220 rad/s, ȟh = 0.27 and Kv = 30.3 (°)/s/V. In order to validate the identification results, the simulation output and experiment output are compared based on the validation data, and shown in Fig.4. The fit is 75.14%.
B C u+ v F D
(2)
where y is the system output, u the system input, v the noise, y-G0 u the residual and A, B, C, D, F are the polynomials with the different orders na, nb, nc, nd, nf. Start with the design of the model error model with a low order model. Finally, for the parameter set na=1, nb=8, nc=5, nd =5, nf =9, we receive an unfalsified error model with tight bounds, nearly over all frequencies, as shown in Fig.5.
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Fig.5
Wang Benyong et al. / Chinese Journal of Aeronautics 23(2010) 240-245
Model error modeling for inner gimbal servo system.
The over-bounding weight Wm can be expressed by Wm =
20(0.2s + 1) 27 s + 1
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output magnitude of controller K, is selected to be 0.1, i.e. Ws=0.1, because the saturation value of the control input voltage to the amplifier is ±10 V. The weighting function Wp is used to reflect the relative importance of various frequency ranges for which performance is desired, and determines the tracking performance boundary of control system. According to the performance requirement for the HFMS servo system and the principle for selection of performance weighting function[9], choose Wp to be 40( s /170 + 1) Wp = (4) s In addition, in Fig.6, e is the tracking error, e1 and e2 are the evaluated errors, u is the control signal, și the command signal, șm the angular position of motor shaft and d, caused by coupling torque, friction torque and others, the disturbance at the plant input.
(3)
The identification procedures for middle gimbal and outer gimbal are similar to that for the inner gimbal just introduced above, and are omitted here. 3. Robust Compound Control
3.1. Design of robust feedback controller In order to attenuate the effect of model uncertainty and external disturbance on the trajectory tracking performance of HFMS servo system, here we introduce robust μ synthesis optimization technique for feedback controller design. Compared with the classical design approaches, the μ synthesis technique provides a flexible systematical framework for feedback controller design and can make the controlled system simultaneously achieve robust stability and robust performance. Before designing robust feedback controller via μ synthesis, describe the performance requirement, parametric uncertainty, high-frequency un-modeled uncertainty and others with weighting functions, and then construct the closed-loop interconnection system with the weighting functions and nominal model. An interconnection diagram for the HFMS inner gimbal closed-loop system, which includes the feedback structure of the plant and controller, and the elements associated with the uncertainty models and performance objectives, is shown in Fig.6. The dashed box represents the true plant, with associated transfer function G. Inside the box are the nominal model of the inner gimbal system dynamics, G0, and two elements, Wm and Δg, which parameterize the uncertainty in the model. The transfer functions G0 and Wm have been obtained with the aforementioned identification procedure. The transfer function Δg is assumed to be stable and unknown, except for the norm condition ||Δg||<1. The weighting function Ws, used to limit the
Fig.6
Closed-loop interconnection structure of inner gimbal single channel system.
Transform the interconnection structure shown in Fig.6 into the basic framework of μ synthesis for the inner gimbal single channel system, namely M-ǻ structure, shown in Fig.7, where Δp is the fictitious uncertainty performance block, the dashed box P represents the transfer function of four-input and four-output open-loop interconnection, and the dashed box M represents the transfer function from [w d și u]T to [z e1 e2 e]T and can be expressed as the lower linear fractional transformation about P and K, i.e. M = Fl (P,K).
Fig.7
M-ǻ structure of μ synthesis for inner gimbal single channel system.
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The design objective of the robust feedback control for HFMS servo system is to design a stabilizing controller K such that for all stable perturbations Δg, with ||Δg||<1, the perturbed closed-loop system remains stable, and the perturbed weighted performance transfer function has H norm not more than 1 for all such perturbations. According to main loop theorem and robust performance theorem[10], this robust design objective exactly fits in with the structured singular value μ framework, and can be solved with D-K iteration[11] program in MATLAB μ analysis and synthesis toolbox[12]. After two D-K iterations, a controller of order 7 is found with the maximum of μ less than 1. The μ plot is shown in Fig.8. The maximum upper bound of μ is 0.985, which guarantees the stability and performance robustness of the closed-loop system with such a control.
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trolled plant almost unchanged, and suppress the external disturbance more effectively than the well-tuned PID controller.
Fig.9
Step responses of disturbance attenuation of inner gimbal single channel system.
3.2. Design of feedforward compensator The block diagram of compound control is shown in Fig.10, where Cff is the feedforward compensator and Δa the additive uncertainty which satisfies |Δa|= |Wm Δg| < |Wm|. The error transfer function ĭe from și to e is
Φe =
Fig.8
μ curve of inner gimbal single channel system.
From the point of implementation in engineering, a controller order reduction is definitely needed. Here is used a balanced realization model reduction method[13-15], which results in an iterative procedure. Start with reducing the controller order to the first order and then carry out the μ test. If the μ test is satisfied, then stop, otherwise reduce the controller to the second order and begin another run. The process goes on until the μ test is satisfied. By such an order reduction procedure, a controller of third order is obtained as follows: K=
3.502 6( s + 17.36)( s 2 + 124.1s + 48 570) s ( s 2 + 249.4 s + 71 250)
(5)
In order to examine the above order reduced D-K iteration robust controller, simulations have been carried out. Fig.9 shows the dynamic responses to the same step disturbance at the plant input. In Fig.9, the curve 1 and curve 2 denote the time responses of the inner gimbal system with the order reduced D-K iteration robust controller at Δg =0 and Δg = 0.9, respectively, and the curves 3 and 4 those with the well-tuned PID controller at the same Δg as the above. From the simulation results, it is clear that, in the presence of model perturbation, the proposed robust controller can keep the responses of the con-
1 − GCff 1 + GK
(6)
It is clear that when Cff = 1/G, the above error transfer function ĭe is 0, which means that with this inverse-feedforward, exact output tracking can be obtained. In practice, the exact plant G may not be known due to modeling errors. Therefore, we replace the true plant G with nominal model G0 to design the feedforward compensator, namely, Cff = 1/G0. Substituting G = G0+ǻa and Cff = 1/G0 into Eq.(6), we obtain
Φe ( jω ) =
Δ ( jω ) Δa ( jω )G0−1 ( jω ) Φe0 ( jω ) (7) = a 1 + G ( jω ) K ( jω ) G0 ( jω )
where Φe0 represents the transfer function from și to e with the feedforward compensator inactive, i.e., Cff = 0. From Eq.(7), it can be seen that |ĭe(jȦ)| < |ĭe0(jȦ)| is obtained when |ǻa(jȦ)| < |G0(jȦ)| or |Wm(jȦ)| < |G0(jȦ)| holds. This means that only if the size of the plant uncertainty is less than the size of the nominal model, the feedforward compensator based on the inverse nominal model can correct tracking error. Generally, the ratio |ǻa(jȦ)|/|G0(jȦ)| is more than 1 at the high frequency. We can shut off the effect of high frequency uncertainty on the tracking performance by introducing non-causal zero phase low-pass filter.
Fig.10
Block diagram of compound control.
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The inverse-feedforward Cff = 1/G0, in fact, is the design method based on the zero-pole cancellation and phase compensation. As the realization problem of digital control is concerned, the system model might have uncancelable zeros which cause unacceptable feedforward compensation. Therefore, we applies the basic idea of zero phase error tracking, which can guarantee the reliability of digital control[16-17], to this inverse-feedforward compensator Cff. Let z transfer function of nominal plant be z − d B ( z −1 ) (8) G0 ( z −1 ) = A( z −1 ) where z−d denotes the d-step pure delay, and B(z−1) and A(z−1) are the polynomials about z−1. Decompose B(z−1) into two parts as B( z −1 ) = B a ( z −1 ) B u ( z −1 ) (9) a −1 where B (z ) contains cancelable zeros of B(z−1) and Bu(z−1) uncancelable zeros of B(z−1). According to Ref.[16], we can obtain the inverse-feedforward compensator based on zero phase error tracking as follows: z d A( z −1 ) B u ( z ) (10) Cff = a −1 u B ( z )( B (1)) 2 where (Bu(1))2 is a scaling factor. The final feedforward compensator is the cascade of the inverse-feedforward compensator based on zero phase error tracking and non-causal zero phase low-pass filter based on |Wm(jȦ)| < |G0(jȦ)|. The procedure to design the compound control for HFMS inner gimbal servo system has been presented. The control design procedures for the other two gimbals are similar and omitted here.
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The experimental results of the inner gimbal single channel are shown in Fig.12, where the curve 1 denotes the desired trajectory, the curve 2 and curve 3 denote the motion trajectories controlled by the proposed approach and well-tuned PID controller, respectively, and the curves 4 and 5 the desired error band. It is seen that the proposed approach possesses better trajectory tracking performance. The results from the experiments on HFMS load coupling suppression are shown in Fig.13, where the curves represent the influences of load coupling on the inner and middle gimbals when HFMS is controlled by the proposed approach and the well-tuned PID controller, respectively. From them, it is clear that the proposed approach exhibits stronger capability of attenuating the load coupling among channels.
4. Experimental
This section is devoted to the experiments on single channel trajectory tracking and load coupling attenuation for evaluating the proposed control strategy. Fig.11 shows the photo of the flight motion simulator with the self-developed motor in the laboratory. The servo valve in the outer gimbal is of MG35 type with a 170 L/min rated flow, the servo valve in the middle gimbal is of YFW06A066AK type with a 66 L/min rated flow, and the servo valve in the inner gimbal is of FF102 type with a 30 L/min rated flow. The experiments run at the oil source pressure of 12 MPa and the oil temperature of 25-35 °C.
Fig.11
Photo of HFMS test prototype.
Fig.12
Experimental results of inner gimbal single channel.
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References [1] [2] [3]
[4] [5] [6] [7] [8]
[9]
[10] [11] [12] [13] Fig.13
Experimental results on HFMS load coupling suppression.
[14]
5. Conclusions [15]
In order to achieve high control performance, in this article we first accurately establish the nominal model and model uncertainty bound for HFMS servo system by system identification, and then propose a robust compound control which is composed of a robust feedback control with μ synthesis and an inverse-feedforward compensation. The results from the simulations and experiments have evidenced that the proposed control strategy not only exhibits better capability of trajectory tracking and attenuating load coupling among channels, but also appears insensitive to external disturbance and strongly robust in the presence of a large model perturbation. Therefore, it stands to reason that the proposed control method completely fits for HFMS hydraulic servo system.
[16] [17]
Liu C. Research on low-speed capability of hydraulic three-axis simulator and its control strategy. PhD thesis, Harbin Institute of Technology, 2003. [in Chinese] Wang P. Theory analysis and experiment research of 3-axis angle vibration hydraulic turntable. MS thesis, Harbin Institute of Technology, 2006. [in Chinese] Xie Y, Mahinda V, Tseng K J, et al. Modeling and robust adaptive control of a 3-axis motion simulator. Proceedings of IEEE Industry Applications Conference. 2001; 1: 553-560. Liu J, Sun F. Nominal model-based sliding mode control with backstepping for 3-axis flight table. Chinese Journal of Aeronautics 2006; 19(1): 65-71. Mei S, Shen T, Liu K. Modern robust control theory and application. 1st ed. Beijing: Tsinghua University Press, 2003; 108-115. [in Chinese] Wang B. Research on three-axis hydraulic simulator control system based on H control theory. PhD thesis, Harbin Institute of Technology, 2008. [in Chinese] Ljung L. System identification theory for the user. 1st ed. Beijing: Tsinghua University Press, 2002; 408-457. Esmaeilsabzali H, Montazeri A, Poshtan J, et al. Robust identification of lightly damped flexible beam using set-membership and model error modeling techniques. Proceeding of the 2006 IEEE international conference on control applications. 2006; 2945-2949. Franchek M A. Slecting the performance weights for the μ and H synthesis methods for SISO regulating systems. ASME Journal of Dynamic Systems Measurement and Control 1996; 118(3): 126-131. Packard A. The complex structured singular value. Automatica 1993; 29(1): 71-109. Stein G, Doyle J C. Beyond singular values and loop shapes. AIAA Journal of Guidance and Control 1991; 14(1): 5-16. Balas G J, Doyle J C, Glover K, et al. μ-analysis and synthesis toolbox. 1st ed. Natick: The MathWorks Inc, 2001; 601-639. Glover K. All optimal Hankel-norm approximations of linear multivariable systems and their L error bounds. International Journal of Control 1984; 39: 1115-1193. Moore B C. Principal components analysis in linear systems: Controllability, observability and model reduction. IEEE Transactions on Automatic Control 1981; 26(1): 17-31. Zhou K, Doyle J C, Glover K. Robust and optimal control. New Jersey: Prentice Hall, 1996; 155-172. Tomizuka M. Zero phase error tracking algorithm for digital control. ASME Journal of Dynamic Systems Measurement and Control 1987; 109(1): 65-68. Tsao T C, Tomizuka M. Adaptive zero phase error tracking algorithm for digital control. ASME Journal of Dynamic Systems Measurement and Control 1987; 109(4): 349-354.
Biography: Wang Benyong Born in 1968, he received Ph.D. degree from Harbin Institute of Technology in 2008. Now, he is a teacher in School of Mechanical Engineering, Heilongjiang Institute of Science and Technology. His main research interest is hydraulic servo control. E-mail: [email protected]
Journal of Aeronautics
Chinese Journal of Aeronautics 23(2010) 240-245
www.elsevier.com/locate/cja
Compound Control for Hydraulic Flight Motion Simulator Wang Benyonga,*, Dong Yanliangb, Zhao Kedingb a
School of Mechanical Engineering, Heilongjiang Institute of Science and Technology, Harbin 150027, China b
School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China Received 20 February 2009; accepted 30 April 2009
Abstract The design of a compound control is presented for the servo system of hydraulic flight motion simulator, which suffers from highly nonlinear dynamics, large parameter time-variation and severe load coupling among channels. The compound control is composed of a robust feedback controller and a feedforward compensator. The design aim is to achieve high tracking performance even in the presence of considerable uncertainty, external disturbance and load coupling among channels. Toward this aim the feedback controller for rejecting perturbation and disturbance is designed by using μ synthesis optimization technique and the feedforward compensator for compensating time lag of dynamic system is established based on the basic idea of zero phase error tracking. To validate the proposed control strategy, simulations and experiments are implemented, and show that the resulting system is highly robust against model perturbation and possesses excellent capability of suppressing the load coupling and improving the tracking performance. Keywords: hydraulic flight motion simulator; system identification; compound control; μ synthesis; feedforward compensation
1. Introduction1 Hydraulic flight motion simulator (HFMS) is a three-degree-of-freedom complex high-accuracy mechanism as shown in Fig.1. HFMS has been widely used in flight motion simulation of various aircraft because it possesses greater power to weight ratio, higher stiffness and less motion error than its electrical rival[1].
1, 8, 11, 15—Servo motor; 2, 7, 10, 14—Angular sensor; 3, 6, 9, 13—Electro-hydraulic servo valve; 4—Middle gimbal; 5—Inner gimbal; 12—Outer gimbal; 16—Base
Fig.1
Schematic structure of a HFMS.
*Corresponding author. Tel.: +86-451-88036252. E-mail address: [email protected] Foundation item: National “211 Project” Foundation of China 1000-9361/$ - see front matter © 2010 Elsevier Ltd. All rights reserved. doi: 10.1016/S1000-9361(09)60211-9
Although HFMS has many distinct advantages, the characteristics of highly nonlinear dynamics, large parameter time-variation and severe load coupling disturbance among channels[2] make various controller design techniques, such as proportion integral derivative (PID), lead-lag compensation, adaptive control[3], sliding model control[4] and others, not be suitable for such high-accuracy plant. The μ synthesis optimization technique[5] can excellently solve this problem due to its extremely effective robustness and enables the threechannel controllers of HFMS to be designed separately. Considering the characteristics of HFMS and the requirement for high tracking performance even in the presence of considerable uncertainty, external disturbance and load coupling among channels, a compound control for HFMS electro-hydraulic servo system is presented in this article. The compound control consists of a robust feedback controller and a feedforward compensator. The feedback controller for attenuating perturbation and disturbance is designed by using μ synthesis optimization technique. The feedforward compensator for compensating time lag of dynamic system is established based on the basic idea of zero phase error tracking. The simulation and experiment results show that the proposed compound control for HFMS servo system is highly robust against large perturbation and extremely effective for suppressing the
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severe load-coupling disturbance among channels and improving system trajectory tracking performance. 2. Modeling The single channel system model for HFMS can be written as[6]
G = G0 + Wm Δg =
Kv
+ Wm Δg (1) §s · 2ξ h s¨ 2 + s + 1¸ © ωh ωh ¹ where G is the real system model, G0 the nominal model, Ȧh the hydraulic undamped natural frequency, ȟh the hydraulic damping ratio, Kv the open-loop gain, s the Laplace operator, Wm Δg the additive error of system model, Wm the weighting function reflecting the amount of uncertainty in the model and Δg denotes the normalized uncertainty satisfying ||Δg||<1. The modeling is accomplished by using system identification[7]. To collect the sufficiently informative data for model identification, the persistently exciting input is chosen to be a pseudo-random binary signal (PRBS) which has the clock time of 0.009 s, the period length of 2 046 and shift between −0.5 V and 0.5 V. A part of the PRBS is shown in Fig.2. The identification procedure is illustrated with the HFMS inner gimbal servo system. 2
Fig.3
Output signals of inner gimbal single channel system.
Fig.4
Examination of inner gimbal system model.
2.2. Estimation of model uncertainty Here the additive error Wm Δg, namely, model uncertainty, is modeled by model error modeling technique based on prediction error algorithm[8]. The dynamic model of Wm Δg can be expressed by A( y − G0u ) =
Fig.2
A part of PRBS for system identification.
2.1. Parameter identification of nominal model As the first step, the above PRBS is input to the inner gimbal single channel system of HFMS, and the output data are recorded and preprocessed as shown in Fig.3. Then the preprocessed data are split into two halves, the first to be used for estimation, and the second one for validation. Based on the estimation data the best nominal model can be computed with prediction error identification algorithm. The obtained identification parameters for inner gimbal servo system are Ȧh = 220 rad/s, ȟh = 0.27 and Kv = 30.3 (°)/s/V. In order to validate the identification results, the simulation output and experiment output are compared based on the validation data, and shown in Fig.4. The fit is 75.14%.
B C u+ v F D
(2)
where y is the system output, u the system input, v the noise, y-G0 u the residual and A, B, C, D, F are the polynomials with the different orders na, nb, nc, nd, nf. Start with the design of the model error model with a low order model. Finally, for the parameter set na=1, nb=8, nc=5, nd =5, nf =9, we receive an unfalsified error model with tight bounds, nearly over all frequencies, as shown in Fig.5.
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Fig.5
Wang Benyong et al. / Chinese Journal of Aeronautics 23(2010) 240-245
Model error modeling for inner gimbal servo system.
The over-bounding weight Wm can be expressed by Wm =
20(0.2s + 1) 27 s + 1
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output magnitude of controller K, is selected to be 0.1, i.e. Ws=0.1, because the saturation value of the control input voltage to the amplifier is ±10 V. The weighting function Wp is used to reflect the relative importance of various frequency ranges for which performance is desired, and determines the tracking performance boundary of control system. According to the performance requirement for the HFMS servo system and the principle for selection of performance weighting function[9], choose Wp to be 40( s /170 + 1) Wp = (4) s In addition, in Fig.6, e is the tracking error, e1 and e2 are the evaluated errors, u is the control signal, și the command signal, șm the angular position of motor shaft and d, caused by coupling torque, friction torque and others, the disturbance at the plant input.
(3)
The identification procedures for middle gimbal and outer gimbal are similar to that for the inner gimbal just introduced above, and are omitted here. 3. Robust Compound Control
3.1. Design of robust feedback controller In order to attenuate the effect of model uncertainty and external disturbance on the trajectory tracking performance of HFMS servo system, here we introduce robust μ synthesis optimization technique for feedback controller design. Compared with the classical design approaches, the μ synthesis technique provides a flexible systematical framework for feedback controller design and can make the controlled system simultaneously achieve robust stability and robust performance. Before designing robust feedback controller via μ synthesis, describe the performance requirement, parametric uncertainty, high-frequency un-modeled uncertainty and others with weighting functions, and then construct the closed-loop interconnection system with the weighting functions and nominal model. An interconnection diagram for the HFMS inner gimbal closed-loop system, which includes the feedback structure of the plant and controller, and the elements associated with the uncertainty models and performance objectives, is shown in Fig.6. The dashed box represents the true plant, with associated transfer function G. Inside the box are the nominal model of the inner gimbal system dynamics, G0, and two elements, Wm and Δg, which parameterize the uncertainty in the model. The transfer functions G0 and Wm have been obtained with the aforementioned identification procedure. The transfer function Δg is assumed to be stable and unknown, except for the norm condition ||Δg||<1. The weighting function Ws, used to limit the
Fig.6
Closed-loop interconnection structure of inner gimbal single channel system.
Transform the interconnection structure shown in Fig.6 into the basic framework of μ synthesis for the inner gimbal single channel system, namely M-ǻ structure, shown in Fig.7, where Δp is the fictitious uncertainty performance block, the dashed box P represents the transfer function of four-input and four-output open-loop interconnection, and the dashed box M represents the transfer function from [w d și u]T to [z e1 e2 e]T and can be expressed as the lower linear fractional transformation about P and K, i.e. M = Fl (P,K).
Fig.7
M-ǻ structure of μ synthesis for inner gimbal single channel system.
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The design objective of the robust feedback control for HFMS servo system is to design a stabilizing controller K such that for all stable perturbations Δg, with ||Δg||<1, the perturbed closed-loop system remains stable, and the perturbed weighted performance transfer function has H norm not more than 1 for all such perturbations. According to main loop theorem and robust performance theorem[10], this robust design objective exactly fits in with the structured singular value μ framework, and can be solved with D-K iteration[11] program in MATLAB μ analysis and synthesis toolbox[12]. After two D-K iterations, a controller of order 7 is found with the maximum of μ less than 1. The μ plot is shown in Fig.8. The maximum upper bound of μ is 0.985, which guarantees the stability and performance robustness of the closed-loop system with such a control.
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trolled plant almost unchanged, and suppress the external disturbance more effectively than the well-tuned PID controller.
Fig.9
Step responses of disturbance attenuation of inner gimbal single channel system.
3.2. Design of feedforward compensator The block diagram of compound control is shown in Fig.10, where Cff is the feedforward compensator and Δa the additive uncertainty which satisfies |Δa|= |Wm Δg| < |Wm|. The error transfer function ĭe from și to e is
Φe =
Fig.8
μ curve of inner gimbal single channel system.
From the point of implementation in engineering, a controller order reduction is definitely needed. Here is used a balanced realization model reduction method[13-15], which results in an iterative procedure. Start with reducing the controller order to the first order and then carry out the μ test. If the μ test is satisfied, then stop, otherwise reduce the controller to the second order and begin another run. The process goes on until the μ test is satisfied. By such an order reduction procedure, a controller of third order is obtained as follows: K=
3.502 6( s + 17.36)( s 2 + 124.1s + 48 570) s ( s 2 + 249.4 s + 71 250)
(5)
In order to examine the above order reduced D-K iteration robust controller, simulations have been carried out. Fig.9 shows the dynamic responses to the same step disturbance at the plant input. In Fig.9, the curve 1 and curve 2 denote the time responses of the inner gimbal system with the order reduced D-K iteration robust controller at Δg =0 and Δg = 0.9, respectively, and the curves 3 and 4 those with the well-tuned PID controller at the same Δg as the above. From the simulation results, it is clear that, in the presence of model perturbation, the proposed robust controller can keep the responses of the con-
1 − GCff 1 + GK
(6)
It is clear that when Cff = 1/G, the above error transfer function ĭe is 0, which means that with this inverse-feedforward, exact output tracking can be obtained. In practice, the exact plant G may not be known due to modeling errors. Therefore, we replace the true plant G with nominal model G0 to design the feedforward compensator, namely, Cff = 1/G0. Substituting G = G0+ǻa and Cff = 1/G0 into Eq.(6), we obtain
Φe ( jω ) =
Δ ( jω ) Δa ( jω )G0−1 ( jω ) Φe0 ( jω ) (7) = a 1 + G ( jω ) K ( jω ) G0 ( jω )
where Φe0 represents the transfer function from și to e with the feedforward compensator inactive, i.e., Cff = 0. From Eq.(7), it can be seen that |ĭe(jȦ)| < |ĭe0(jȦ)| is obtained when |ǻa(jȦ)| < |G0(jȦ)| or |Wm(jȦ)| < |G0(jȦ)| holds. This means that only if the size of the plant uncertainty is less than the size of the nominal model, the feedforward compensator based on the inverse nominal model can correct tracking error. Generally, the ratio |ǻa(jȦ)|/|G0(jȦ)| is more than 1 at the high frequency. We can shut off the effect of high frequency uncertainty on the tracking performance by introducing non-causal zero phase low-pass filter.
Fig.10
Block diagram of compound control.
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The inverse-feedforward Cff = 1/G0, in fact, is the design method based on the zero-pole cancellation and phase compensation. As the realization problem of digital control is concerned, the system model might have uncancelable zeros which cause unacceptable feedforward compensation. Therefore, we applies the basic idea of zero phase error tracking, which can guarantee the reliability of digital control[16-17], to this inverse-feedforward compensator Cff. Let z transfer function of nominal plant be z − d B ( z −1 ) (8) G0 ( z −1 ) = A( z −1 ) where z−d denotes the d-step pure delay, and B(z−1) and A(z−1) are the polynomials about z−1. Decompose B(z−1) into two parts as B( z −1 ) = B a ( z −1 ) B u ( z −1 ) (9) a −1 where B (z ) contains cancelable zeros of B(z−1) and Bu(z−1) uncancelable zeros of B(z−1). According to Ref.[16], we can obtain the inverse-feedforward compensator based on zero phase error tracking as follows: z d A( z −1 ) B u ( z ) (10) Cff = a −1 u B ( z )( B (1)) 2 where (Bu(1))2 is a scaling factor. The final feedforward compensator is the cascade of the inverse-feedforward compensator based on zero phase error tracking and non-causal zero phase low-pass filter based on |Wm(jȦ)| < |G0(jȦ)|. The procedure to design the compound control for HFMS inner gimbal servo system has been presented. The control design procedures for the other two gimbals are similar and omitted here.
No.2
The experimental results of the inner gimbal single channel are shown in Fig.12, where the curve 1 denotes the desired trajectory, the curve 2 and curve 3 denote the motion trajectories controlled by the proposed approach and well-tuned PID controller, respectively, and the curves 4 and 5 the desired error band. It is seen that the proposed approach possesses better trajectory tracking performance. The results from the experiments on HFMS load coupling suppression are shown in Fig.13, where the curves represent the influences of load coupling on the inner and middle gimbals when HFMS is controlled by the proposed approach and the well-tuned PID controller, respectively. From them, it is clear that the proposed approach exhibits stronger capability of attenuating the load coupling among channels.
4. Experimental
This section is devoted to the experiments on single channel trajectory tracking and load coupling attenuation for evaluating the proposed control strategy. Fig.11 shows the photo of the flight motion simulator with the self-developed motor in the laboratory. The servo valve in the outer gimbal is of MG35 type with a 170 L/min rated flow, the servo valve in the middle gimbal is of YFW06A066AK type with a 66 L/min rated flow, and the servo valve in the inner gimbal is of FF102 type with a 30 L/min rated flow. The experiments run at the oil source pressure of 12 MPa and the oil temperature of 25-35 °C.
Fig.11
Photo of HFMS test prototype.
Fig.12
Experimental results of inner gimbal single channel.
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References [1] [2] [3]
[4] [5] [6] [7] [8]
[9]
[10] [11] [12] [13] Fig.13
Experimental results on HFMS load coupling suppression.
[14]
5. Conclusions [15]
In order to achieve high control performance, in this article we first accurately establish the nominal model and model uncertainty bound for HFMS servo system by system identification, and then propose a robust compound control which is composed of a robust feedback control with μ synthesis and an inverse-feedforward compensation. The results from the simulations and experiments have evidenced that the proposed control strategy not only exhibits better capability of trajectory tracking and attenuating load coupling among channels, but also appears insensitive to external disturbance and strongly robust in the presence of a large model perturbation. Therefore, it stands to reason that the proposed control method completely fits for HFMS hydraulic servo system.
[16] [17]
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Biography: Wang Benyong Born in 1968, he received Ph.D. degree from Harbin Institute of Technology in 2008. Now, he is a teacher in School of Mechanical Engineering, Heilongjiang Institute of Science and Technology. His main research interest is hydraulic servo control. E-mail: [email protected]