导航制导TuA2-02

Jianwei Cheng School of Aerospace Science Beijing institute of technology Beijing, China cjw03919@bit.edu.cn Xiaoxian Yao School of Aerospace Science Beijing institute of technology Beijing, China yxx11@bit.edu.cn Zhiyuan Yu China Academy of Launch Vehicle Technology Beijing, China happytimmyu@yahoo.cn Abstract—Fuzzy sliding mode controller based on trending law for electric actuator is proposed in this paper. The sliding mode is used to enhance the robustness of the system, the trending law is used to improve the system dynamic characteristics of reaching the sliding surface, and the fuzzy tuning schemes are employed to reduce the chattering problem around the sliding surface. The integral sliding mode controller with exponential reaching law was developed. And the parameters of the exponential trending law were configured by fuzzy control rule. The method is applied to the control of an electric actuator. And the results of the simulation verify the validity of the proposed approaches. I. INTRODUCTION ncertainty in dynamical non-linear systems which suffer from external disturbances and parameters variation is considered as a difficult challenge in designing the control system. Sliding mode control has been suggested as an approach for the control of systems with nonlinearities, uncertain dynamics and bounded input disturbances for the last two decades and was widely recognized as a potential approach to this problem. The most distinguished feature of the sliding mode control technique is its ability to provide fast error convergence and strong robustness for control systems in the sense that the closed loop systems are completely insensitive to nonlinearities, uncertain dynamics, uncertain system parameters and bounded input disturbances in the sliding mode. However, chattering phenomena will take place when implementing a sliding mode control as there are discontinuous control actions. Chattering problem is an obvious drawback of sliding mode in application. The reasons are as follows. In a real system, the sliding mode variable structure around the sliding surface accompanied with high frequency chattering as there are time delay switch, space delay switch, system inertia, measure error and other factors. The chattering not only affects the control accuracy, increase the energy consumptions, but also may excite the high-frequency un-modeled dynamics parts. One common way to eliminate this drawback is to introduce the trend law to sliding mode. This method can not only improve the system dynamic characters of reaching the sliding surface but also eliminate high frequency chattering of the control signal through adjusting the parametersε and k . Furthermore, ε may bring chattering problems when it’s too big. Fuzzy tuning schemes are employed in order to solve this problem effectively. The fuzzy tuning law is proposed by analyzing the relationship between the switching function and the parameters of exponential turning law. The parameters ε and k are the output of the fuzzy controller. In this paper, an integral sliding mode controller is developed to minish the electric actuator system error. The exponential turning law is employed to improve the dynamic characters of reaching the sliding surface. Moreover, Fuzzy U Fuzzy Sliding Mode Controller Based on Trending Law for Electric Actuator 978-1-4244-6044-1/10/$26.00 ©2010 IEEE eq su u u= + control and sliding mode are combined together to eliminate the chattering problem. II. SLIDING MODE BASED ON EXPONENTIAL TRENDING LAW The mathematic model of a typical electric actuator system is given by Eq. (1) as follows. It is obvious that the system is a second derivative system. 4 2 ( ) ( ) v Ks U s T s s δ = + (1) We design a sliding mode scheme for the following n degree-of-freedom linear SISO system ( , )x Ax Bu Df x t= + +� (2) Where, nx∈ℜ is the state vector of the system, u is the input of the system, ( , )f x t is the external disturbances of the system, 0D is the maximal value of the disturbances. Usually we define the switching function as 1 1 2 2( ) n ns x c x c x c x= + + +" (3) When 1nc = , and Eq. (3) can been rewrite as 1 1 2 2( ) ns x c x c x x= + + +" (4) In order to reduce the system error, a new state vector is introduced, and it is defined as 0 1x x=� , which means 0 1( ) ( )dx t x t t= ∫ (5) Thus, extending the system to 1n + degree-of-freedom and the system can be described as ( , )Ex A x bu f x t= + +� (6) Where, 1nx +∈ℜ . The switching function is 0 0 1 1 2 2( ) ns x Cx c x c x c x x= = + + + +" (7) Use Ackermann-formula to choose , 0,1, 2, , 1ic i n= −" Then the Ackermann-formula is expressed as ( )T EC e P A= (8) [ ] 110, ,0,1 , , ,T nE Ee b A b A b −−⎡ ⎤= ⎣ ⎦" " (9) ( )( ) ( )( )0 1 1( ) n nP λ λ λ λ λ λ λ λ λ−= − − − −" 2 2 0 1 2 1 n n nP P P Pλ λ λ λ−−= + + + + +" (10) 1 0 [1,0, ,0]0 I E I n A A AA × ⎡ ⎤= =⎢ ⎥⎣ ⎦ " Where, ( )P λ is the characteristic polynomial of the system, 0 1 nλ λ λ"， ，， are the expectant eigenvalues of the switch surface. The control function u is make up of two parts, and is shown as (11) Where, equ is the equivalent control, the function of this term is to force the system state to slide on the sliding surface, and su is the switch control part which is used to meet the needs of compensating the uncertainty, non-linear and other unideal factors of the system. Consider Eq. (11) and the exponential turning law together, we can get a new expression as ( ) ( )1 0( )sgn( )u CB CAx ks D sε−= − + + + (12) Where, the equivalent control part and the switch control part can been written as follows. ( ) 1equ CB CAx−= − (13) ( ) [ ]1 0( ) sgn( )su CB ks D sε−= − + + (14) Where 0D can be ignored, and use sgn( )sε to compensate it. III. FUZZY CONTROL The fuzzy controller use s and s� as system input,ε and k as system output. The scheme of fuzzy sliding mode controller can be shown as figure 1. We define that 1 2n ns k s ksε σ ε σ= − = −� (15) 1ns sα= (16) 2n ns sα=� � (17) Where, 1 2 1 2, , ,σ σ α α are quantization factors, and they are all positive. Moreover, we define the fuzzy sets of input ,n ns s� and output nε , k as: { }3 2 1 0 1 2 3, , , , , ,A A A A A A A− − − −� � � � � � � { }3 2 1 0 1 2 3, , , , , ,B B B B B B B− − −� � � � � � � { }3 2 1 0 1 2 3, , , , , ,C C C C C C C− − −� � � � � � � and{ }1 2 3, ,D D D� � � . Choose seven linguistic variables for ,n ns s� and nε . The linguistic variables are defined as follows: e , kε ,s s� r δd SMC FC Electric Actuator Fig.1. Scheme of fuzzy sliding mode controller 978-1-4244-6044-1/10/$26.00 ©2010 IEEE {Positive Big (PB), Positive Medium (PM), Positive Small (PS), Zero (ZE), Negative Small (NS), Negative Medium (NM), Negative Big (NB)}. Where, the corresponding linguistic values of , , , ( 3, 2, 1,0,1, 2,3)i i iA B C i = − − −� �� are NB, NM, NS, ZE, PS, PM, PB. Choose three linguistic variables of nk as {Low (L), Medium (M), High (H)}, and the corresponding linguistic values of ( 1,2,3)iD i =� are L, M and H. Use triangle of membership functions, and they are shown as below in figure 2 to 5. μ s Fig.2. Membership functions of ns μ � Fig.3. Membership functions of ns� μ nε Fig.4. Membership functions of nε Fig.5. Membership functions of nk Then we can establish the fuzzy control rules which are made up of a series of “IF -THEN” expressions. Notice that the sliding mode can be divided into four states according to figure 6: (I) 0 & 0s s> >� ; (II) 0 & 0s s> <� ; (III) 0& 0s s< >� ; (IV) 0 & 0s s< <� . Choose the parameters ε and k according to the distance between system state and the switching surface and the tending speed to the switching surface. Fig.6. The fuzzification of switching function for a fuzzy sliding mode controller If the system is located in (I) 0 & 0s s> >� and { }n ns s PB= =� which means the system state is positive far from the sliding surface and is trending to go away from the surface in high speed. Thus, the control function should have positive big trending speed and high control force. The other situations can be analyzed in the same way. And the fuzzy control rule can be shown in table 1 as below. TABLE 1. Rule table for the /n nkε ns� ns NB NM NS ZE PS PM PB NB NB/H NB/H NB/H NB/M NM/L NS/L ZE/L NM NB/H NB/H NB/M NM/M NS/L ZE/L PS/L NS NB/H NB/H NM/H NS/M ZE/L PS/L PM/L ZE NB/M NM/M NS/M ZE/M PS/M PM/M PB/M PS NM/L NS/L ZE/L PS/M PM/H PB/H PB/H PM NS/L ZE/L PS/L PM/M PB/H PB/H PB/H PB ZE/L PS/L PM/L PB/M PB/H PB/H PB/H The fuzzy control rule can be described in linguistic form as follow: lR : If ns is iA� , and ns� is jB� then nε is ijC� , and nk is ijD� , ( , 3, 2, , 2,3)i j = − − " . Define the control rule 1R A B C= × →� �� and 2R A B D= × →� � � , so the reasoning result can be shown as follows. ( ) ( ) ( ) ( )1 2 1 2U A B R R A B R A B R C D ⎡ ⎤ ⎡ ⎤= × = × ×⎣ ⎦ ⎣ ⎦ = � � �� � �D ∪ D ∪ D � �∪ ( ) ( ) ( ) ( ) C A A C B B C D A A D B B D ⎧ ⎡ ⎤= → →⎪ ⎣ ⎦⎨ ⎡ ⎤= → →⎪ ⎣ ⎦⎩ � � � � �� �D ∩ D � �� � � � �D ∩ D So the membership functions can be shown as ( ) ( ) BC A C D B DA z z μ ω ω ω μ ω ω ω = ∧ ∧⎧⎨ = ∧ ∧⎩ � � � � � � � � (18) 978-1-4244-6044-1/10/$26.00 ©2010 IEEE Where ( )ω • is the maximum of the membership functions, and z is the independent variable of the membership functions. Clarify the output of the fuzzy controller through choosing center average defuzzification. 3 3 3 3 3 3 3 3 ( ) ( ) ( ) ( ) C n C D n D z z dz z dz z z dz k z dz με μ μ μ − − − − ⎧⎪ =⎪⎪⎨⎪⎪ =⎪⎩ ∫ ∫ ∫ ∫ � � � � (19) IV. SIMULATION& ELECTRIC ACTUATOR SYSTEM The transfer function of the electric actuator system can be rewritten in form of differential equation as follows: 4( ) ( ) ( )VT t t K u tδ δ+ =�� � (20) Define the error of the system as i f i fe u u u K δ= − = − (21) Thus, 1 ( ) 1 ( ) 1 ( )i i f i f f u e u e K u e K K δ δ δ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ = − = − = −�� �� � � � �� (22) Put equ. (20) and equ. (22) together, we have a new equation 2 ( ) ( )e a e bu t F t+ = − +�� � (23) Where, ( )F t is the generalized disturbance of the system, and can be defined as 2 1( ) ( ) ( )i iF t u t a u t bf= + −� � (24) Where 2 1 v a T = , 4f v K K b T = . The maximum and minimum of the system disturbances can be estimated as below ( )F t F≤ Define 1 2 1,x e x x= = � , furthermore a new state variable: 0x e=� . Then the error state equation can be extended as 0 1 1 2 2 2 2 ( ) ( ) x x x x x a x bu t F t =⎧⎪ =⎨⎪ = − − +⎩ � � � (25) The simulation results of unit step responses of the three different controllers can be seen in figure.7. Fig.7. Simulation result of unit step response Moreover, the time domain characteristics can be shown in table.2 as below. TABLE 2. Time domain characteristics Rise time(ms) Adjust time(ms) Overshoot PID 19.41 31.78 0.1% SMC 14.19 26.0 0.1% FSMC 12.25 23.0 0.1% It can be see through simulation that all of the three controllers have small overshoot, but for response time, the SMC is better than PID controller and the FSMC is better than SMC, which verify the dynamic adjusting of the fuzzy controller. -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 e (a) Phase track of SMC -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -160 -140 -120 -100 -80 -60 -40 -20 0 20 e de (b) Phase track of FSMC Fig.8. Simulation result of phase tracks The FSMC is not only faster but also can weaken chattering compared with SMC. As shown in figure 8, (a) is the phase track of SMC. And the chattering amplitude is 978-1-4244-6044-1/10/$26.00 ©2010 IEEE about 0.4. (b) is the phase track of FSMC. And the chattering amplitude is about 0.02 which is much smaller than 0.4. It is obvious that the fuzzy control eliminates the chattering problem of SMC. V. CONCLUSION In this paper a fuzzy sliding mode controller based on trending law for electric actuator is studied. An integral sliding mode controller is conducted to reduce the electric actuator system error. The fuzzy control is developed to configure the parameters of sliding mode control based on exponential turning law in order to eliminate the chattering problem. The fuzzy rules are constructed according to the characters of the switching function and the sliding surface. The simulation and analytic results verify the superiority of the FSMC and its efficiency of weakening the chattering. REFERENCE [1] V. A. Utkin, “Variable structure systems with sliding mode,” IEEE Trans. Automat. Contr., vol. AC-22, no. 2, 1977, pp. 212-222. [2] J. Rolf, "Adaptive Controller of Robot Manipulator Motion.'' IEEE Trans. on Robotic and Automation, Vol. 6, No. 2. 1990. [3] Q. P. Ha, and M. Negnevitsky, “A robust modal controller with fuzzy tuning for multimass electromechanical systems,” Proc. IEEE 3rd Australian and New-Zealand Conf. On intelligent information systems, Perth, Australia, 1995, pp. 412-219. [4] W.A. Deabes, Fyez Areed. Fuzzy Sliding Motion Controller for Six-Degree-of-Freedom Robotic Manipulator. IEEE 41st Southeastern Symposium on System Theory University of Tennessee Space Institute Tullahoma, TN, USA, March 15-17, 2009, pp.148-152. [5] Xiaojiang Zhang, Zhihong Man. A' New Fuzzy Sliding Mode Control Scheme. IEEE Proceedings of the 3d World Congress on Intelligent Control and Automation. Hefei, P.R. China. June 28-July 2,2000, pp.1692-1696 [6] JIANG K, ZHANG J G, CHEN Z M. A new approach for the sliding mode control based on fuzzy reaching law[C]. Proc of the 4th World Congress on Intelligent Control and Automation. Shanghai, China: Press of University of Science and Technology of China, 2002, 6: 656-660. [7] Jurgen Ackermann and Vadim I. Utkin．Sliding Mode Control Design Based on Ackermann’s Formula[C]．Lake Buena Vista, FL: 1994 [8] X.J. Liu, P. Guan, and J.Z. Liu. Fuzzy Sliding Mode Attitude Control of Satellite. Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005, 1970-1975. [9] Hai-Ping Pang, Cheng-Ju Liu and Wei Zhang. Sliding Mode Fuzzy Control with Application to Electrical Servo Drive. Proceedings of the Sixth International Conference on Intelligent Systems Design and Applications (ISDA'06). 2006. [10] J. C. Wu and T. S. Liu. A Sliding-Mode Approach to Fuzzy Control Design. IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 4, NO. 2, MARCH 1996,pp.141-151. 978-1-4244-6044-1/10/$26.00 ©2010 IEEE