Nội dung

TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K1- 2015 Control of active suspension system using and adaptive robust controls  Trong Hieu Bui  Quoc Toan Truong H University of Technology, VNU- HCM ABSTRACT: This paper presents a control of active suspension system for quarter-car model with two-degree-of-freedom using H  and nonlinear adaptive robust control method. Suspension dynamics is linear and treated by H  method which guarantees the robustness of closed loop system under the presence of uncertainties and minimizes the effect of road disturbance to system. An Adaptive Robust Control (ARC) technique is used to design a force controller such that it is robust against actuator uncertainties. Simulation results are given for both frequency and time domains to verify the effectiveness of the designed controllers. Keywords: Active suspension, Hydraulic actuator, H  control, Adaptive robust control. 1. INTRODUCTION Automotive suspension systems have been developed from the begin time of car industrial with a simple passive mechanism to the present with a very high level of sophistication. Suspensions incorporating active components are studied to improve the overall ride performances of automotive vehicle in recent years. Active suspension must provide a trade-off between several competing objectives: passenger comfort, small suspension stroke for packing and small tire deflection for vehicle handling. In the early studies, linear model of suspension are used with the assumption of ideal force actuator. The most applicable force actuator using in practice is hydraulic actuator that has a high non-linearity characteristic. Hence to solve completely problem, recently studies consider to the dynamics and the non-linearity of hydraulic actuator [2,7,9]. This paper presents a control of active suspension system for quarter-car model with two-degree-of-freedom by using H  and nonlinear adaptive robust control method. The system is divided into two parts: the linear part is whole system except actuator and nonlinear part is hydraulic actuator. The linear part is treated using H  control method that guarantees the robustness of closed loop system under the presence of uncertainties and minimizes the effect of disturbance. The variations of system parameters are solved by multiplicative uncertainty model. In hydraulic actuator, there are some unknown factors such as bulk modulus of hydraulic fluid that has strong effect to actuator dynamics. Hence, the nonlinear adaptive control is suitable for designing actuator controller. This paper applied the ARC technique to design a the controller robust against actuator uncertainties[3,4]. The error between desired acting force calculated from H  controller and actual force generated by hydraulic actuator is considered as the disturbance to the linear system. Simulations have been done in both frequency and time domains to verify the effectiveness of the designed controllers. Trang 5 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 18, No.K1- 2015 2. SYSTEM MODELING The scheme of suspension system and hydraulic actuator used in this paper is described in Fig. 1. zs ks F zu xvalve ms ms bs acting force pr ps pr F mu spool valve mu kt hydraulic cylinder zr a. Quarter-car model b. Hydraulic actuator x6  xvalve : position of valve from its closed position. The governing dynamic equations of suspension system including hydraulic actuator can be presented as the following[9] x1  x2  x4 1  k s x1  bs ( x2  x4 )  x5  ms (2) x3  x4  zr (3) x2  x4  1 k s x1  bs ( x2  x4 )  kt x 3  x5  mu (4) x5   x5   f A2 ( x2  x4 )  Fig.1 Suspension system and actuator Define parameters as the follows  A Ps A  sgn( x6 ) x5 x6 ms : sprung mass  bs : damping coefficient ks : spring stiffness coefficient    f Cd w f 1 /  kt : tire stiffness coefficient F zs : active force    f Ctm (6) where, : displacement of the car body z u : displacement of wheel : displacement of road Assume that the spring stiffness coefficient and tire stiffness coefficient are linear in their operation range; the tire does not leave the ground; and z s and z u are measured from the static equilibrium point. From the scheme of the system model in the Fig. 1, the state variables are chosen as follows x1  z s  zu : suspension deflection x2  zs : velocity of car body x3  zu  z r : tire deflection x4  zu x5  F Trang 6 (5) 1 x6  ( x6  u ) mu : unsprung mass zr (1) : velocity of wheel : active force  f  4 e / Vt A : piston area Ps : supply pressure of the fluid C d : discharge coefficient wf : spool valve area gradient  : hydraulic fluid density Ctm : total leakage coefficient of the piston e : effective bulk modulus Vt : total actuator volume  u : time constant : input to servo-valve Equations (1)-(4) represent the quarter-car dynamics and equations (5)-(6) drive the hydraulic actuator dynamics. TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K1- 2015 3. H CONTROL OF LINEAR PART Let’s define the force error e  x5  x5 where d (7) x5 is actual control force generated d from actuator and x5 is the desired control force which is calculated from H controller. Consider x5 as the control input, the systems (1)-(4) can be rewritten in the form  z xp  Ap x p  B p x5    r  e tire deflection zu  z r . Then three considered transfer functions from disturbance  z r to the acceleration of the sprung mass H A (s) , to the suspension deflection H SD (s) , and to the tire deflection H TD (s) can be derived  ( s )  ( s) Z X s 2  ( s) ( s) Z Z r r H A ( s)  (10) Z s ( s)  Z u ( s) X 1 ( s)  Zr ( s) Zr ( s) (11)  x1  x  xp   2  ,  x3     x4  H TD ( s)  Z u ( s)  Z r ( s) X ( s)  3 Zr ( s) Zr ( s ) (12) The augmented system G(s) for 1 b  s ms 0 bs mu  0   1   m  Bp   s  ,  0   1   mu  0  1  C Tp    0    0  the H SD ( s)  (9) where  0  ks  m s Ap    0  ks  mu measured by the deflection of suspension z s  zu ; and tire load constancy, measured by as the following (8) and the measured output is the velocity of car body yp  Cp xp Three interest performance variables are: body vibration isolation, measured by the sprung mass acceleration  z s ; suspension travel, 0 0 0 k  t mu 1  bs  ms   1  b  s  mu  0  0 1   0 ms   , 0   1 0  1   mu  H control problem is given in the Fig. 2. z  r  w n  e G(s) x2 P(s) u x5 x5 W1 z x2 W2 zu y Hydraulic Actuator x5d K(s) x5d Fig. 2. Configuration of control system The state space expression of the plant P(s) with adding measurement noise n can be written in the following form xp  Ap x p  B p1w  B p 2 x5 z p  C p1 x p  D p11w  D p12 x5 (13) (14) Trang 7 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 18, No.K1- 2015 y p  C p 2 x p  D p 21w  D p 22 x5 (15) The state space expression of the plant G(s) can be written as follows x Ax  B1 w  B2 x5 (16) z  C1 x  D11w  D12 x5 (17) y  C2 x  D21w  D22 x5 (18) where, x  x   p  xw  , z  z p , y  y p , 0  Ap A   BwC p11 Aw   Bp2   B p1  B1   B   2 B D   Bw D p111 ,  w p121 ,  Dw C p11 C1    w D p12 C2  C p 2 0 Bw  0  ,  Dw D p111  Dw D p121 D11   D12      w D p122   w D p112  , D21  D p 21 D22  D p 22 , The H control problem is to find an internal stabilizing controller, K (s) , for the augmented system, G(s) , such that the inf-norm of the closed loop transfer function, given positive scalar Find K ( s) stabilizin g  Tzw  Tzw , is below a  (19) Furthermore, from the small gain theorem the robust stability of the closed loop system under presence of parameter uncertainty is assured if   1 . Here the change of the parameters of the system is treated by multiplicative uncertainty Trang 8 model (s) . It is derived from the nominal plant Pn (s) and the perturbed plant Pp (s) as follows ( s )  Pp ( s) Pn ( s) 1 (20) The weighting is chosen to satisfy  [(s)]  W1 (s) ,  (21) The transfer function from disturbance to the state of the augmented system is 1 Tx zr  sI  [ A  B2 K ( s)C 2 ]1[ B1  B2 K ( s) D21 ]0 0 (22) where K (s) is H  controller. Three transfer functions (10)-(12) become H AC (s)  sE2 H SD (s)  E1 H TD (s)  E3 0Tx zr 0Tx zr 0Tx zr where E1  1 0 0 0, E2  0 1 0 0 , E3  0 0 1 0 4. ADAPTIVE ROBUST CONTROL OF NONLINEAR PART In this part we will derive the controller for hydraulic actuator used in suspension system. The controller is designed based on adaptive robust control technique proposed by Bin Yao [3]. Consider hydraulic actuator dynamic equations (5)-(6). The parameter is considered as unknown parameter  f  4 e / Vt . The main reason for choosing f as unknown factor is that the bulk modulus of hydraulic fluid is known to change dramatically even when there is a small leakage between piston and cylinder. The equation (5) can be written in the form x5   [a1 x5  a2 ( x2  x4 )  a3 Ps A  sgn( x6 ) x5 x6 ]  d (23) TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K1- 2015 a1  Ctm ; where a3  Cd w f A /  ;  and is a2   A2 ; a  unknown (28) d denotes disturbances and parameter; 1 a3   1  a1 x5  a 2 ( x2  x4 )  ˆ ( x5d  k1 z1 )    r   1 z1   M2 (a1 x5  a 2 ( x2  x4 )  4 min a3   11 their extents are known a3 a ) 2       { :  min     max } | d | d M b  a3 Ps A  sgn( x6 ) x5 ˆ   (ˆ)  M   max   min    (24) x5   [a1 x5  a2 ( x2  x4 )  bx6 ]  d  is estimated by adaptation law (25) Define the error variable: 1  0 (26) To find a virtual control law  for  x6 such x5 tracks its desired value x5d using the procedure suggested in [3]. The term b , that representing the nonlinear static gain between the flow rate and the valve opening x6 , is a function x6 and also is non-smooth since x6 appears through a discontinuous function sgn( x6 ) . So a of [3] Define the smooth projection  (ˆ) :    1   max    1  exp  (ˆ   max )       ˆ ˆ  ( )       1  exp  1 (ˆ   )  min     min       The adaptive part (ˆ   max ) (ˆ  [ min , max ]) (ˆ   min ) is given by   a   r (27) a using the following and the robust control are calculated as follows and (30) is a known arbitrary small positive number  11 ,  12 are adjustable small positive numbers. Step 2: To find an actual control law for such that u x6 tracks the desired control function  synthesized in step 1 with a guaranteed transient performance. smooth modification is needed .  ˆ ˆ   1 z1[a1 x5  a2 ( x2  x4 )  a3 a ] , z1  x5  x5d r (29) k1 : tunable parameter Equation (23) becomes part  d M2   12  1 where The adaptive control law can be obtained as the following steps. Step 1: Let’s define The control law 1 Define the error variable z 2  x6   (31) Adaptive robust control law consists of two parts: an adaptive part and a robust control part u  u a  ur (32) The adaptive part and robust control part are calculated as follows ua      k 2 z 2  pe   1 2c   b ˆ  (33) Trang 9 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 18, No.K1- 2015 ur    4b Table 1. Numerical values for simulation z 2 h2 Parameters Values Units where ms 290 kg w 1 b  pe  ˆ 1 z1  bx6  xˆ5  c w2  x5 mu 59 kg bs 1000 Ns/m ks 16812 N/m kt 190000 N/m N/m5 (34) (35) ˆ5  ˆ [a1 x5  a2 ( x2  x4 )  bx6 ] x (36)  ̂  x5  x5 t (37) f 4.515e13  2c   1c  w2 z 2  (38)  1.00  1.545e9 N/(m5/2kg1/2) A 3.35e-4 m2 Ps 10342500 N/m2 c   1c  w1 z1 a1 x5  a2 ( x2  x4 )  a3  (39)   b   w1 z1    a1 x5  a2 ( x2  x4 )  a3  w2  x5 x5  (40) h2  1 2  M2  2 k 2 , w1 , w2 (41) and 2 are arbitrary positive numbers. 5. SIMULATION RESULTS Frequency domain The plot of uncertainties and weighting functions are given in Fig. 3. Figures (4)-(6) show the gain plots for three transfer functions (10)-(12) in cases of passive system, active system with desired force and actual force input. As shown in the figures, the designed nonlinear ARC controller can treat the nonlinearity and H keep the The numerical values using in this simulation are given in the Table 1[9]. The weighting function is chosen as The controller is calculated with the value of   0.99 . The road velocity disturbance is assumed to be from road displacement r  0.1sin 2 f t . The parameters of ARC  1  5e6 , k1  150 , k 2  10 ,    0.001 ,  11  5 ,  12  2 ,  2  5 and d M  2 . 20 0  ( j ) for  bs W ( j ) -20 Gain (dB)  3.135s  9.2625  0 W ( s) 0    W ( s)   1  0 . 93 s  29  W   4   0 0 3.5  10   frequency performance well. -40  ( j ) for  ms -60  ( j ) for  ks -80 -100  ( j ) for  kt -120 -140 10 -2 10 -1 10 0 10 1 10 2 Frequency (Hz) controller are chosen to be Trang 10 Fig. 3. Plots of uncertainties and weighting function TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K1- 2015 6 Suspension deflection (mm) 30 20 Gain (dB) 10 0 passive system -10 active system with x5d input -20 ****** -30 active system with x5 input 4 2 0 -2 -4 passive system -6 active system -40 10 0 10 -1 10 1 10 2 -8 Frequency (Hz) 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2 Time (s) Fig. 4. Gain plots for body acceleration transfer function Fig. 8. Suspension deflection with step disturbance -10 -20 6 -30 4 Tire deflection (mm) Gain (dB) -40 -50 -60 passive system -70 active system with x5d input -80 ****** -90 active system with x5 input -100 2 0 -2 -4 passive system -6 active system -110 10 -1 10 0 10 1 10 2 -8 0 Frequency (Hz) 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2 Time (s) Fig. 5. Gain plots for suspension deflection transfer function Fig. 9. Tire deflection with step disturbance -25 1.5 -30 1 Acceleration (m/s2) Gain (dB) -35 -40 -45 passive system -50 active system with x5d input ****** -55 0.5 0 -0.5 active system with x5 input -1 -1.5 passive system active system -60 10 -1 10 0 10 1 -2 10 2 0 Frequency (Hz) 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) Fig. 6. Gain plots for tire deflection transfer function Fig. 10. Acceleration with sine disturbance 2 5 Suspension deflection (mm) 6 passive system Acceleration (m/s2) 4 active system 3 2 1 0 -1 -2 1.5 1 0.5 0 -0.5 -1 -1.5 passive system -3 active system -2 0 -4 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) Time (s) Fig. 11. Suspension deflection with sine disturbance Fig. 7. Acceleration with step disturbance Trang 11 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 18, No.K1- 2015 system in case of sine wave disturbance are given in Figs. (10)-(12). The road amplitude is assumed to be 0.1 m with frequency of 1 Hz . At this frequency, active system reduces considerably the effects of disturbance. 2 Tire deflection (mm) 1.5 1 0.5 0 -0.5 6. CONCLUSION This paper presents a control of active suspension -1 -1.5 passive system active system -2 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) Fig. 12. Tire deflection with sine disturbance Time domain The responses of the system with step and sine wave disturbances are considered. Responses of the system in case of step disturbance are given in Figs. (7)-(9). The step road velocity is of 0.1 m/s. Body acceleration and tire deflection are much reduced but the suspension deflection is higher. Responses of the system using H and nonlinear adaptive robust control method. H  controller achieved the robustness with the presence of parameter uncertainties and minimized the effects of disturbance. The nonlinear ARC controller treats well the non-linearity and the parameter uncertainties of hydraulic actuator. Simulation results show that the designed controller can keep the good performance of H  controller in both frequency and time domains. Điều khiển hệ thống treo chủ động của xe ô tô dùng H  và điều khiển thích nghi bền vững  Trong Hieu Bui  Quoc Toan Truong Trường Đại học Bách khoa, ĐHQG-HCM TÓM TẮT: Điều khiển hệ thống treo chủ động của xe ô tô là một đề tài thú vị trong lĩnh vực nghiên cứu về ô tô. Bài báo này đề xuất phương pháp điều khiển hệ thống treo chủ động bằng lý thuyết H  và điều khiển thích nghi bền vững. Kỹ thuật điều khiển thích nghi bền vững (ARC) được sử dụng để thiết kế bộ điều khiển lực bền vững với các thông số không biết chắc của bộ chấp hành. Kết quả mô phỏng đã thể hiện tính hiệu quả của bộ điều khiển đề nghị. Từ khóa: : Hệ thống treo chủ động, Điều khiển H  , Điều khiển thích nghi bền vững. Trang 12 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K1- 2015 REFERENCES [1]. [2]. [3]. [4]. [5]. T.T. Nguyen, V.G. Nguyen and S.B. Kim, Control of Active Suspension System by Using Hinf Theory, ICASE Transaction on Control, Automation, and System Engineering, Vol. 2, No. 1, pp. 1-6, March, 2000. Takanori Fukao, Arika Yamawaki and Norihiko Adachi, Nonlinear and Hinf Control of Active Suspension Systems with Hydraulic Actuators, Proceeding of the 38th Conference on Decision and Control, pp. 5125-5128, Phoenix, Arizona USA, December 1999. Bin Yao, George T.C. Chiu, John T. Reedy, Nonlinear Adaptive Robust Control of OneDOF Electro_Hydraulic Servo Systems, ASME International Mechanical Engineering Congress and Exposition, pp. 191-197, 1997. [6]. T.T. Nguyen, Control of Active Suspension System by Using H  Theory, MS. Thesis, PKNU, Korea, June 1998. [7]. Jung-Shan Lin and Ioannis Kanellakopoulos, Nonlinear Design of Active Suspensions, IEEE Control Systems Magazine, Vol. 17, No. 3, pp. 45-59, June 1997. [8]. Kemin Zhou, John C. Doyle and Keith Glover, Robust and Optimal Control, Prentice Hall, Inc., 1996. [9]. Andrew Alleyne and J. Karl Hedrick, Nonlinear Adaptive Control of Active Suspensions, IEEE Transaction on Control Systems Technology, Vol. 3, No. 1, pp. 94101, March 1995. [10]. M. Krstic, I. Kanellakopoulos and P. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc., 1995. Bin Yao and M. Tomizuka, Adaptive Robust Control of SISO nonlinear systems in a semi-strict feedback form, Automatica, vol. 33, no. 5, pp. 893-900, 1997. [11]. M. Yamashita, K. Fujimori, K. Hayakawa Supavut Chantranuwathana and Huei Peng, Adaptive Robust Control for Active Suspensions, Proceeding of the American Control Conference, pp. 1702-1706, San Diego, California USA, June 1999. [12]. Jean-Jacques E. Slotine and Weiping Li, and H. Kimura, Application of Hinf Control to Active Suspension System, Automatica, Vol. 30, No. 11, pp. 1717-1729, 1994. Applied Non-linear Control, Prentice-Hall, Inc., 1991. Trang 13