- Email: [email protected]
Zero phase error control based on neural compensation for flight simulator servo system*
Journal of Systems Engineering and Electronics, Vol. 17, No. 4, 2006, pp. 793-191
Zero phase error control based on neural compensation for flight simulator servo system* Liu Jinkun' , He Peng2& Er Lianjie' 1.School of Automation Science & Electrical Engineering, Beijing Univ. of Aeronautics and Astronautics,Beijing 100083, P.R.China; 2.Computing Lad,Oxford Univ.Parks Road, Oxford,OX13QD U.K. (Received October 13,2005) Abstract: Using the future desired input value, zero phase error controller enables the overall system's frequency response exhibit zero phase shift for all frequencies and a small gain error at low frequency range, and based on this, a new algorithm is presented to design the feedforward controller. However, zero phase error controller is only suitable for certain linear system. To reduce the tracking error and improve robustness, the design of the proposed feedforward controller uses a neural compensation based on diagonal recurrent neural network. Simulation and real-time control results for flight simulator servo system show the effectiveness of the proposed approach. Keywords: zero phase error, servo system, neural network, robust control, flight simulator.
1. INTRODUCTION In high precision tracking applications of mechanical systems, feedforward controllers are often used to enhance tracking performance, and they are generally prefilter for the desired trajectory to compensate for the dynamic lag of the closed loop and to extend the tracking bandwidth. Because of the rapid growth of computer technology, digital motion control system has become more and more popular, and feedforward controller design in discrete time domain attracts much attention. The classical feedforward controller is based on pole-zero cancellation algorithm, which makes the overall transfer h c t i o n to be equal be unity, thus, perfect tracking can be achieved. However, when the closed loop system is of nonmimal phase, the cancellation of unstable zeros will result in the instability the feedforward controller. To handle the unstable zeros, Tomuzuka"' proposed the well known zero phase error tracking control algorithm (ZPETC). By using the preview information of the desired output, ZPETC cancels the phase lags caused by unstable zeros. In recent years, relative researches on ZPETC-based feedforward controller have mainly focus on three aspects: the improvement of gain characteristics of ZPETC""'], the optimal des- ign of ZPETC[6-71and the enhanc-ement of robus- tness of ZPETC to parameter
However, in some applications, such as flight simulator servo system, the preview information of the signal is not known in advance"'], and only the present and previous steps of the desired output are available, moreover, the transfer function is often uncertain. For such situation, ZPETC-based feedforward controller is not applicable. This paper presents a new ZPETC-based feedforward controller algorithm that adopts neural compensation, and the preview information of the desired output is not necessary.
2. DESCRIPTION OF THE SYSTEM A typical servo system, can be described as a second
rank transfer function as follows 1 b G, (s) z z -= (1) Js2 + gs s(s + a ) Where J is equivalent moment of inertia of the frame, and g is equivalent coefficient. The transfer fbnction (1) can be written as a discrete position state equation as follows X ( k + 1) = A , X ( k )+ B,u(k)
Y ( k + 1) = C , X ( k + 1) + D,
(2)
Moreover, the state Eq.(2) can be transformed to discrete error state equation using sampling time t, as follows X,(k + 1) = A,X,(k) + Be@) + f ( k ) (31 where f ( k ) = R ( k + l ) - A , R ( k ) , A, = A , , Be =-B, ,
X,(k)= [e(k) de(k)lT,R ( k + 1) represents the ideal
The project was supported by Aeronautics Foundation of China (00E51022).
Liu Jinkun , He Peng & Er Lianjie
794
input state at time k + l , R ( k ) = [ r ( k ) i ( k ) l T , e(k)
Supposing
and de(k) represents system error and system error
(7)
change at time k respectively, r ( k ) is ideal From theorem 1, we can get
position signal at time k ,e ( k ) = r ( k )- y (k).
3.
LG(e-JmT) =0
CONTROL SYSTEM DESIGN
Consider a SISO system as Fig. 1 .
Fig1
?C)
Closed loop system G,(z-')
The transfer fhction of closed system G,(z-') is
G,(z-')=
z - d N , (z-')N,(z-1) D( z-' )
E
R
(8)
G(l) = 1 (9) From Eq. (8) an important conclusion can be drawn: if the desired output of y d ( k ) , y d ( k + l ). ., . , y d (k + d + rn) is known before- hand, by using of ZPETC, the phase shift from y,(k) to y ( k ) is zero for all fiequencies, and the gain error is very small at low fiequency range. Assuming that the ZPE control input is F , ( k ) , the ZPE controller can be described as
3.1 ZPETC Design
plant
,Vw
ej ( k ) = Fr ( k ) - ~
(4)
( k )
u p ( k ) = k p * e j ( k ) = k p. ( F r ( k ) - y ( k ) )
(10)
where kp > 0. Where f d represents a d -step delay, and N,(z-') includes the unstable and light damped zeros, N,(Z-') includes the stable and well damped zeros. Note that all zeros of closed loop system (4) outside of the unit cycle are included in Nu(z-') . Theorem 1 Supposing H(z-') = Nu( z ) N , ( z - ' ) , two conclusions can be proved"] as follows (1) (2)
LH(e-JmT) = 0 ,V w E R ILH(e-JmT)/2 = Re2[N,(e-JfoT)]+Im2[N,(e-JmT)],Vw E R
Let's denote the ZPETC controller for such a system as(shown in Fig.2)
Fig.2
A ZPE control structure
Therefore, we obtain the practical position as
Y ( k )= F(z-')Gc(z-')r(k+ m + d)=
In practical control, especially in flight simulator servo system, the value of desired output is often not known beforehand, therefore if ZPETC is used in such servo system, good tracking performance will not be guaranteed. Moreover, the robustness of ZPETC is not good. 3.2 D R " Neural Compensation Recently, the need for high-precision servo system has considerably increased in many applications, especially in flight simulator servo system. In such servo systems, the moment of inertia and the load often change, and nonlinear frictional dynamics can reduce the tracking performance. Moreover, these effects are generally dependent on the hardware architecture of the motors and may also change with time. Diagonal Recurrent Neural Network ( D R " ) is a recurrent neural network["], that can be used to approach any continuous function with high precision [12]. In this paper D R " is used to compensate ZPE controller, and to improve position tacking precision and robustness performance. The D R " is a three-layers structure, its hidden layer is recurrent layer. The structure of D R " is depicted in Fig.3. In the D R " neural network, I = [ I l , 1 2 , , . . , I , , ] is the input vector, Zi(k) is the ith neural net of
Zero phase error control based on neural compensationfor fright simulator servo system
795
value is adopted as follows
-s(k)Be( 2)xj ( k ) $(k) = w; (k - 1) + qoAw; ( k )+
a (w;( k - 1) - w; ( k - 2))
(17)
F i g 3 The DRNN Neural Network Structure
input layer, the jth net output of recurrent layer is X j ( k ) ,S j ( k ) is total input of jth recurrent neural net, f(.) is Sigmoid function and O(k) is output of DRNN. The overall algorithm of D R " is O(k)=
c wpxj
( k ) ,xj ( k )= f(Sj (k))
i
s/( k ) = w; x j( k - 1) +
cwlf li
(k)
( 11)
i
Where W Dand W o are weight vector of the recurrent layer and output layer respectively, both are dimension of nh and W' is weight matrix of input layer. The weight matrix W' is W L [ @ i w,l i ... *:
wfi iw;] ***
(12)
Where Wf is the weight value of input layer, whose dimension is n , , n, and nh are number of neural nets of input layer and recurrent layer respectively. The output of DRNN is un ( k ) = O(k)
(13) 3.3
The position tracking error change are e(k) - e(k - 1) de ( k ) = ts Where r, is sampling time. Sliding mode surface is defined as s ( k ) = c x e(k)+ de(k)= C, . X ,
Where qo , q D,.q, are learning rate of input layer, recurrent layer and output layer respectively, and a is inertial factor. Overall Structure of Control System
According to above discussion, the control system structure is described as shown in Fig.4. + dkl
(14)
where C, = [c 13 ,c >O. In the D R " neural network, two kind of input are chosen as x(1) = s(k),x(2) = s ( k )- s(k - 1) (15) The learning index is chosen as 1 E ( k ) = -s(k)* (16) 2 According to steepest gradient method, considering as(k) dun ( k )
z Be(2)
, the
learning algorithm of weight
F i g 4 The control system structure
The controller can be described as u(k) = u p ( k )+ u, ( k ) (20) Where u p @ ) represents ZPE control input given by Eq. (10) and u , ( k ) represents neural compensation given by Eq. (13).
Liu Jinkun , He Peng & Er Lianjie
796
4. EXPERIMENTS
4.2 Real-time control
Simulation and real-time control are carried out on a three-axis flight simulator system. The sampling time is chosen as ts=l ms=O.OOl s, each axis of the flight simulator is driven by a DC torque motor, and an inductosyn is used to measure the angle position, the resolution is 0.000 1. In this paper, we only consider the middle fiame of flight simulator servo system. 4.1 Simulation Example
The transfer function is tested by HP frequency instrument and is approached as a second rank as follows 1 G,(s) =Js2+qs For position tracking, choosing the desired trajectory as sine signal r(k)= A sin (27d7t) Where t = k x t, , A is signal magnitude and F is signal fiequency. In this paper, consideringJ and q are timevarying parameters, we choose k = 2 000, A=0.50, F=3.0, 1 25 0.5sin (10nt). J=-+O.Ssin (10n;t), q=-+ 133 133 The position tracking results can be seen in Fig.5, hich indicates ZPE controller has bad robustness. The position tracking can be observed using ZPE controller with neural compensation, as shown in Fig.6, the result indicates that the proposed method has better performance. 1 05 0
$ 0 -0.5
0.5
I ;Is
15
I
2
Fig.5 ZPE position tracking
Moreover, we use ZPE controller and ZPE controller with neural compensation in real-time control application. The position tracking results are given in Fig.7 and Fig.8. The results indicate that higher position precision can be achieved using the ZPE controller with D R " compensation compared with the use of only the ZPE controller, and system robustness can be considerably improved the use of only. 1 05 0
$
2
0 -05
0.5
I
IS
I
2
rls
Fig.7 Real-time position tracking with NN compensation based ZPE controller
r1s
Fig.8 Real-time position tracking with ZPE controller only
5. CONCLUSIONS In this paper, a novel method is proposed for designing the feedforward controller in the case that the preview information of the desired output is not known and the transfer function is uncertain. In the proposed approach, the neural compensation is used to improve system robustness and position tracking precision. The simulation and real-time control results show that the controller has high robustness and high position tracking precision performance, which is very suitable for high precision servo system with uncertainties. The stability of the closed system will be dealt with in the fiture studies.
' I
0.5 0
L?
$ 0
REFERENCES
-0.5
ds
Fig.6 ZPE position tracking with NN compensation
[l] Tomizuka M. Zero phase error tracking algorithm for digital control. ASME Journal of Dynamic Systems, Measurement, and Control, 1987,109:65-68. [2] Haack B, Tomizuka M. The effect of adding zeros to feed-
Zero phase error control based on neural compensationfor fright simulator servo system forward controllers. ASME Journal of Dynamic Systems, Measurement, andControl,1991, 113: 6-10. [3]Torfs D.Extended bandwidth zero phase error tracking control of nonminimal phase systems. ASME Journal of Dynamic Systems, Measurement, and Control, 1992,114: 347-35 1. [4]Menq C H,Chen J J. Precision tracking control of discrete time nonminimum-phase systems. ASME Journal of Dynamic Systems, Measurement, and Control, 1993,115:238-
245. [5] Xia J Z, Menq C H. Precision tracking control of nonminimum phase systems with zero phase error. International Journal of Contro1,1995,61:791-807. [6]Funahashi Y, Yamada M. Zero phase error tracking controllers with optimal gain characteristics. ASME Journal of Dynamic Systems, Measurement, and Control, 1993, 1 15:
311-318. [7]Tsao T C. Optimal feed-forward digital tracking controller design. ASME Journal of Dynamic Systems, Measur- ernent, and Control, 1994, 116:583-591. [8]Tsao T C, Tomizuka M. Adaptive zero phase error tracking algorithm for digital control. ASME Journal of Dynamic Systems, Measurement, and Control, 1987, 109:349-354. [9]Yeh S S , Hsu P L. An optimal and adaptive design of the feedforward motion controller. IEEE Transactions on Mechotronics, 1999,4:428-438. [lo] Liu J K, Liu Q, Er L J. Zero phase error real time control for flight simulator servo system. Chinese Journal of Mechanical Engineering, 2004,17(1): 132-135. [ll]Ku C C. Kwang YL. Diagonal recurrent neural networks for dynamic system control. IEEE Transactions on Neural Networks,1995,6:144-156. [ 121 Sabanovic A, Jezernik K, Rodic M. Neural network application in sliding mode control systems. 1996 IEEE Worhhop on Variable Structure Systems, 1996:143- 147.
797
Liu Jinkun was born in 1965. He received B.S., M.S. and Ph.D. degrees in department of automatic control from Northeastern University, China in 1989, 1994 and 1997, respectively. After a two-year post-doctor at automatic department in Zhejiang University from 1997 to 1999 and half-year research associate at Hongkong Technology University in 1999, he joined department of automatic control, Beijing University of Aeronautics and Astronautics in 1999, as an associate professor. His research concerns motion control, intelligent control and robust control. E-mail: [email protected] He Peng received Undergraduate degree from the University of Hong Kong in 2006. Now, he is a M.Sc student in the Computing Laboratory in Oxford University. His early research interests were in the fields of PDA hardware architecture design, Win 32 Sound API programming and lighting tracing stimulation. Currently, his main research interests are focused on 3D terrain real-time display remotely environmental impact surveillance, neural network and computer control. E-mail: [email protected] Er Lianjie was born in Tianjin City, China in Aug., 1938. He graduated from department of automatic control, Beijing University of Aeronautics and Astronautics in 1960. As a professor, his research concerns high precision motion control and variable structure control. He has published about twenty papers.
Zero phase error control based on neural compensation for flight simulator servo system* Liu Jinkun' , He Peng2& Er Lianjie' 1.School of Automation Science & Electrical Engineering, Beijing Univ. of Aeronautics and Astronautics,Beijing 100083, P.R.China; 2.Computing Lad,Oxford Univ.Parks Road, Oxford,OX13QD U.K. (Received October 13,2005) Abstract: Using the future desired input value, zero phase error controller enables the overall system's frequency response exhibit zero phase shift for all frequencies and a small gain error at low frequency range, and based on this, a new algorithm is presented to design the feedforward controller. However, zero phase error controller is only suitable for certain linear system. To reduce the tracking error and improve robustness, the design of the proposed feedforward controller uses a neural compensation based on diagonal recurrent neural network. Simulation and real-time control results for flight simulator servo system show the effectiveness of the proposed approach. Keywords: zero phase error, servo system, neural network, robust control, flight simulator.
1. INTRODUCTION In high precision tracking applications of mechanical systems, feedforward controllers are often used to enhance tracking performance, and they are generally prefilter for the desired trajectory to compensate for the dynamic lag of the closed loop and to extend the tracking bandwidth. Because of the rapid growth of computer technology, digital motion control system has become more and more popular, and feedforward controller design in discrete time domain attracts much attention. The classical feedforward controller is based on pole-zero cancellation algorithm, which makes the overall transfer h c t i o n to be equal be unity, thus, perfect tracking can be achieved. However, when the closed loop system is of nonmimal phase, the cancellation of unstable zeros will result in the instability the feedforward controller. To handle the unstable zeros, Tomuzuka"' proposed the well known zero phase error tracking control algorithm (ZPETC). By using the preview information of the desired output, ZPETC cancels the phase lags caused by unstable zeros. In recent years, relative researches on ZPETC-based feedforward controller have mainly focus on three aspects: the improvement of gain characteristics of ZPETC""'], the optimal des- ign of ZPETC[6-71and the enhanc-ement of robus- tness of ZPETC to parameter
However, in some applications, such as flight simulator servo system, the preview information of the signal is not known in advance"'], and only the present and previous steps of the desired output are available, moreover, the transfer function is often uncertain. For such situation, ZPETC-based feedforward controller is not applicable. This paper presents a new ZPETC-based feedforward controller algorithm that adopts neural compensation, and the preview information of the desired output is not necessary.
2. DESCRIPTION OF THE SYSTEM A typical servo system, can be described as a second
rank transfer function as follows 1 b G, (s) z z -= (1) Js2 + gs s(s + a ) Where J is equivalent moment of inertia of the frame, and g is equivalent coefficient. The transfer fbnction (1) can be written as a discrete position state equation as follows X ( k + 1) = A , X ( k )+ B,u(k)
Y ( k + 1) = C , X ( k + 1) + D,
(2)
Moreover, the state Eq.(2) can be transformed to discrete error state equation using sampling time t, as follows X,(k + 1) = A,X,(k) + Be@) + f ( k ) (31 where f ( k ) = R ( k + l ) - A , R ( k ) , A, = A , , Be =-B, ,
X,(k)= [e(k) de(k)lT,R ( k + 1) represents the ideal
The project was supported by Aeronautics Foundation of China (00E51022).
Liu Jinkun , He Peng & Er Lianjie
794
input state at time k + l , R ( k ) = [ r ( k ) i ( k ) l T , e(k)
Supposing
and de(k) represents system error and system error
(7)
change at time k respectively, r ( k ) is ideal From theorem 1, we can get
position signal at time k ,e ( k ) = r ( k )- y (k).
3.
LG(e-JmT) =0
CONTROL SYSTEM DESIGN
Consider a SISO system as Fig. 1 .
Fig1
?C)
Closed loop system G,(z-')
The transfer fhction of closed system G,(z-') is
G,(z-')=
z - d N , (z-')N,(z-1) D( z-' )
E
R
(8)
G(l) = 1 (9) From Eq. (8) an important conclusion can be drawn: if the desired output of y d ( k ) , y d ( k + l ). ., . , y d (k + d + rn) is known before- hand, by using of ZPETC, the phase shift from y,(k) to y ( k ) is zero for all fiequencies, and the gain error is very small at low fiequency range. Assuming that the ZPE control input is F , ( k ) , the ZPE controller can be described as
3.1 ZPETC Design
plant
,Vw
ej ( k ) = Fr ( k ) - ~
(4)
( k )
u p ( k ) = k p * e j ( k ) = k p. ( F r ( k ) - y ( k ) )
(10)
where kp > 0. Where f d represents a d -step delay, and N,(z-') includes the unstable and light damped zeros, N,(Z-') includes the stable and well damped zeros. Note that all zeros of closed loop system (4) outside of the unit cycle are included in Nu(z-') . Theorem 1 Supposing H(z-') = Nu( z ) N , ( z - ' ) , two conclusions can be proved"] as follows (1) (2)
LH(e-JmT) = 0 ,V w E R ILH(e-JmT)/2 = Re2[N,(e-JfoT)]+Im2[N,(e-JmT)],Vw E R
Let's denote the ZPETC controller for such a system as(shown in Fig.2)
Fig.2
A ZPE control structure
Therefore, we obtain the practical position as
Y ( k )= F(z-')Gc(z-')r(k+ m + d)=
In practical control, especially in flight simulator servo system, the value of desired output is often not known beforehand, therefore if ZPETC is used in such servo system, good tracking performance will not be guaranteed. Moreover, the robustness of ZPETC is not good. 3.2 D R " Neural Compensation Recently, the need for high-precision servo system has considerably increased in many applications, especially in flight simulator servo system. In such servo systems, the moment of inertia and the load often change, and nonlinear frictional dynamics can reduce the tracking performance. Moreover, these effects are generally dependent on the hardware architecture of the motors and may also change with time. Diagonal Recurrent Neural Network ( D R " ) is a recurrent neural network["], that can be used to approach any continuous function with high precision [12]. In this paper D R " is used to compensate ZPE controller, and to improve position tacking precision and robustness performance. The D R " is a three-layers structure, its hidden layer is recurrent layer. The structure of D R " is depicted in Fig.3. In the D R " neural network, I = [ I l , 1 2 , , . . , I , , ] is the input vector, Zi(k) is the ith neural net of
Zero phase error control based on neural compensationfor fright simulator servo system
795
value is adopted as follows
-s(k)Be( 2)xj ( k ) $(k) = w; (k - 1) + qoAw; ( k )+
a (w;( k - 1) - w; ( k - 2))
(17)
F i g 3 The DRNN Neural Network Structure
input layer, the jth net output of recurrent layer is X j ( k ) ,S j ( k ) is total input of jth recurrent neural net, f(.) is Sigmoid function and O(k) is output of DRNN. The overall algorithm of D R " is O(k)=
c wpxj
( k ) ,xj ( k )= f(Sj (k))
i
s/( k ) = w; x j( k - 1) +
cwlf li
(k)
( 11)
i
Where W Dand W o are weight vector of the recurrent layer and output layer respectively, both are dimension of nh and W' is weight matrix of input layer. The weight matrix W' is W L [ @ i w,l i ... *:
wfi iw;] ***
(12)
Where Wf is the weight value of input layer, whose dimension is n , , n, and nh are number of neural nets of input layer and recurrent layer respectively. The output of DRNN is un ( k ) = O(k)
(13) 3.3
The position tracking error change are e(k) - e(k - 1) de ( k ) = ts Where r, is sampling time. Sliding mode surface is defined as s ( k ) = c x e(k)+ de(k)= C, . X ,
Where qo , q D,.q, are learning rate of input layer, recurrent layer and output layer respectively, and a is inertial factor. Overall Structure of Control System
According to above discussion, the control system structure is described as shown in Fig.4. + dkl
(14)
where C, = [c 13 ,c >O. In the D R " neural network, two kind of input are chosen as x(1) = s(k),x(2) = s ( k )- s(k - 1) (15) The learning index is chosen as 1 E ( k ) = -s(k)* (16) 2 According to steepest gradient method, considering as(k) dun ( k )
z Be(2)
, the
learning algorithm of weight
F i g 4 The control system structure
The controller can be described as u(k) = u p ( k )+ u, ( k ) (20) Where u p @ ) represents ZPE control input given by Eq. (10) and u , ( k ) represents neural compensation given by Eq. (13).
Liu Jinkun , He Peng & Er Lianjie
796
4. EXPERIMENTS
4.2 Real-time control
Simulation and real-time control are carried out on a three-axis flight simulator system. The sampling time is chosen as ts=l ms=O.OOl s, each axis of the flight simulator is driven by a DC torque motor, and an inductosyn is used to measure the angle position, the resolution is 0.000 1. In this paper, we only consider the middle fiame of flight simulator servo system. 4.1 Simulation Example
The transfer function is tested by HP frequency instrument and is approached as a second rank as follows 1 G,(s) =Js2+qs For position tracking, choosing the desired trajectory as sine signal r(k)= A sin (27d7t) Where t = k x t, , A is signal magnitude and F is signal fiequency. In this paper, consideringJ and q are timevarying parameters, we choose k = 2 000, A=0.50, F=3.0, 1 25 0.5sin (10nt). J=-+O.Ssin (10n;t), q=-+ 133 133 The position tracking results can be seen in Fig.5, hich indicates ZPE controller has bad robustness. The position tracking can be observed using ZPE controller with neural compensation, as shown in Fig.6, the result indicates that the proposed method has better performance. 1 05 0
$ 0 -0.5
0.5
I ;Is
15
I
2
Fig.5 ZPE position tracking
Moreover, we use ZPE controller and ZPE controller with neural compensation in real-time control application. The position tracking results are given in Fig.7 and Fig.8. The results indicate that higher position precision can be achieved using the ZPE controller with D R " compensation compared with the use of only the ZPE controller, and system robustness can be considerably improved the use of only. 1 05 0
$
2
0 -05
0.5
I
IS
I
2
rls
Fig.7 Real-time position tracking with NN compensation based ZPE controller
r1s
Fig.8 Real-time position tracking with ZPE controller only
5. CONCLUSIONS In this paper, a novel method is proposed for designing the feedforward controller in the case that the preview information of the desired output is not known and the transfer function is uncertain. In the proposed approach, the neural compensation is used to improve system robustness and position tracking precision. The simulation and real-time control results show that the controller has high robustness and high position tracking precision performance, which is very suitable for high precision servo system with uncertainties. The stability of the closed system will be dealt with in the fiture studies.
' I
0.5 0
L?
$ 0
REFERENCES
-0.5
ds
Fig.6 ZPE position tracking with NN compensation
[l] Tomizuka M. Zero phase error tracking algorithm for digital control. ASME Journal of Dynamic Systems, Measurement, and Control, 1987,109:65-68. [2] Haack B, Tomizuka M. The effect of adding zeros to feed-
Zero phase error control based on neural compensationfor fright simulator servo system forward controllers. ASME Journal of Dynamic Systems, Measurement, andControl,1991, 113: 6-10. [3]Torfs D.Extended bandwidth zero phase error tracking control of nonminimal phase systems. ASME Journal of Dynamic Systems, Measurement, and Control, 1992,114: 347-35 1. [4]Menq C H,Chen J J. Precision tracking control of discrete time nonminimum-phase systems. ASME Journal of Dynamic Systems, Measurement, and Control, 1993,115:238-
245. [5] Xia J Z, Menq C H. Precision tracking control of nonminimum phase systems with zero phase error. International Journal of Contro1,1995,61:791-807. [6]Funahashi Y, Yamada M. Zero phase error tracking controllers with optimal gain characteristics. ASME Journal of Dynamic Systems, Measurement, and Control, 1993, 1 15:
311-318. [7]Tsao T C. Optimal feed-forward digital tracking controller design. ASME Journal of Dynamic Systems, Measur- ernent, and Control, 1994, 116:583-591. [8]Tsao T C, Tomizuka M. Adaptive zero phase error tracking algorithm for digital control. ASME Journal of Dynamic Systems, Measurement, and Control, 1987, 109:349-354. [9]Yeh S S , Hsu P L. An optimal and adaptive design of the feedforward motion controller. IEEE Transactions on Mechotronics, 1999,4:428-438. [lo] Liu J K, Liu Q, Er L J. Zero phase error real time control for flight simulator servo system. Chinese Journal of Mechanical Engineering, 2004,17(1): 132-135. [ll]Ku C C. Kwang YL. Diagonal recurrent neural networks for dynamic system control. IEEE Transactions on Neural Networks,1995,6:144-156. [ 121 Sabanovic A, Jezernik K, Rodic M. Neural network application in sliding mode control systems. 1996 IEEE Worhhop on Variable Structure Systems, 1996:143- 147.
797
Liu Jinkun was born in 1965. He received B.S., M.S. and Ph.D. degrees in department of automatic control from Northeastern University, China in 1989, 1994 and 1997, respectively. After a two-year post-doctor at automatic department in Zhejiang University from 1997 to 1999 and half-year research associate at Hongkong Technology University in 1999, he joined department of automatic control, Beijing University of Aeronautics and Astronautics in 1999, as an associate professor. His research concerns motion control, intelligent control and robust control. E-mail: [email protected] He Peng received Undergraduate degree from the University of Hong Kong in 2006. Now, he is a M.Sc student in the Computing Laboratory in Oxford University. His early research interests were in the fields of PDA hardware architecture design, Win 32 Sound API programming and lighting tracing stimulation. Currently, his main research interests are focused on 3D terrain real-time display remotely environmental impact surveillance, neural network and computer control. E-mail: [email protected] Er Lianjie was born in Tianjin City, China in Aug., 1938. He graduated from department of automatic control, Beijing University of Aeronautics and Astronautics in 1960. As a professor, his research concerns high precision motion control and variable structure control. He has published about twenty papers.