Control of active suspension system using H and adaptive robust controls

TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K1- 2015 Control of active suspension system using H and adaptive robust controls  Trong Hieu Bui  Quoc Toan Truong University of Technology, VNU- HCM ABSTRACT: This paper presents a control of uncertainties and minimizes the effect of active suspension system for quarter-car road disturbance to system. An Adaptive model with two-degree-of-freedom using Robust Control (ARC) technique is used H  and nonlinear adaptive robust to design a force controller such that it is control method. Suspension dynamics is robust against actuator uncertainties. linear and treated by method which Simulation results are given for both guarantees the robustness of closed frequency and time domains to verify the loop system under the presence of effectiveness of the designed controllers. Keywords: Active suspension, Hydraulic actuator, control, Adaptive robust control. 1. INTRODUCTION Automotive suspension systems have been system is divided into two parts: the linear part is developed from the begin time of car industrial whole system except actuator and nonlinear part with a simple passive mechanism to the present is hydraulic actuator. The linear part is treated with a very high level of sophistication. using control method that guarantees the Suspensions incorporating active components are robustness of closed loop system under the studied to improve the overall ride performances of automotive vehicle in recent years. Active presence of uncertainties and minimizes the suspension must provide a trade-off between effect of disturbance. The variations of system parameters are solved by multiplicative several competing objectives: passenger comfort, uncertainty model. In hydraulic actuator, there small suspension stroke for packing and small tire are some unknown factors such as bulk modulus deflection for vehicle handling. In the early of hydraulic fluid that has strong effect to studies, linear model of suspension are used with the assumption of ideal force actuator. The most actuator dynamics. Hence, the nonlinear adaptive applicable force actuator using in practice is control is suitable for designing actuator controller. This paper applied the ARC technique hydraulic actuator that has a high non-linearity to design a the controller robust against actuator characteristic. Hence to solve completely uncertainties[3,4]. The error between desired acting problem, recently studies consider to the dynamics and the non-linearity of hydraulic force calculated from controller and actual [2,7,9] actuator . force generated by hydraulic actuator is This paper presents a control of active considered as the disturbance to the linear suspension system for quarter-car model with system. Simulations have been done in both frequency and time domains to verify the two-degree-of-freedom by using and effectiveness of the designed controllers. nonlinear adaptive robust control method. The Trang 5 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 18, No.K1- 2015 2. SYSTEM MODELING x6  xvalve : position of valve from its The scheme of suspension system and closed position. hydraulic actuator used in this paper is described The governing dynamic equations of suspension in Fig. 1. system including hydraulic actuator can be presented as the following[9] z x s ms valve ms p x x  x bs r 1 2 4 (1) ks F acting ps force F p 1 z r  u m x2   ks x1 bs (x2  x4 )  x5  u spool valve m s (2) m k u t hydraulic cylinder x  x  z 3 4 r (3) zr 1 a. Quarter-car model b. Hydraulic actuator x  k x  b (x  x )  k x x  4 m s 1 s 2 4 t 3 5 u (4) Fig.1 Suspension system and actuator 2 x5   x5  f A (x2  x4 )  Define parameters as the follows  A Ps Asgn(x6 )x5 x6 ms : sprung mass (5) 1 mu : unsprung mass x6  (x6  u)  (6) bs : damping coefficient where, k : spring stiffness coefficient s    C w 1/  f d f kt : tire stiffness coefficient    f Ctm F : active force  f  4e /Vt zs : displacement of the car body A : piston area zu : displacement of wheel P s : supply pressure of the fluid zr : displacement of road C Assume that the spring stiffness coefficient d : discharge coefficient and tire stiffness coefficient are linear in their w operation range; the tire does not leave the f : spool valve area gradient ground; and z and z are measured from the s u  : hydraulic fluid density static equilibrium point. From the scheme of the C system model in the Fig. 1, the state variables are tm : total leakage coefficient of the piston chosen as follows  e : effective bulk modulus x1  zs  zu : suspension deflection Vt x2  zs : velocity of car : total actuator volume body  : time constant u x3  zu  zr : tire deflection : input to servo-valve Equations (1)-(4) represent the quarter-car  : velocity of wheel x4  zu dynamics and equations (5)-(6) drive the hydraulic actuator dynamics. x5  F : active force Trang 6 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K1- 2015 Three interest performance variables are: 3. H CONTROL OF LINEAR PART body vibration isolation, measured by the sprung Let’s define the force error mass acceleration z; suspension travel, e  x  x d s 5 5 (7) measured by the deflection of suspension zs  zu ; and tire load constancy, measured by where x5 is actual control force generated d the tire deflection zu  zr . Then three from actuator and x5 is the desired control force considered transfer functions from disturbance which is calculated from H  controller. Consider zr to the acceleration of the sprung mass x as the control input, the systems (1)-(4) can 5 H (s) , to the suspension deflection H (s) , be rewritten in the form A SD and to the tire deflection H (s) can be derived z TD x  A x  B x   r as the following p p p p 5  e    (8) Z(s) X (s) H (s)  s  2 and the measured output is the velocity of car A Z(s) Z(s) r r body (10) y p  C p x p (9) Z s (s)  Zu (s) X1 (s) H SD(s)   where Z(s) Z(s) r r (11) x1    Zu (s)  Z r (s) X 3 (s) x HTD (s)    2    x p  , Z r (s) Z r (s) x3  (12)   x4  The augmented system G(s) for  0 1 0 1  control problem is given in the Fig. 2.  k b b   s  s 0 s   G(s) ms ms ms x A  zr 2 p    z 0 0 0 1  W1 x2   w n  ks bs kt bs   u    e P(s) W z   2 u  mu mu mu mu  x5 y  0   0 0   1   1  x5 Hydraulic 0 xd  m   m  Actuator 5 s s d K(s) B p    ,     , x  0  1 0  5 1 1    0    m   m   u   u  Fig. 2. Configuration of control system 0 The state space expression of the plant P(s)   with adding measurement noise n can be written T 1 C    in the following form p 0   xp  Ap xp  Bp1w Bp2 x5 0 (13) z  C x  D w D x p p1 p p11 p12 5 (14) Trang 7 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 18, No.K1- 2015 y  C x  D w D x model (s) . It is derived from the nominal plant p p2 p p21 p22 5 (15) P (s) and the perturbed plant P (s) as follows The state space expression of the plant G(s) n p can be written as follows P (s) (s)  p 1 x Ax  B1w B2 x5 P (s) (16) n (20) z  C x  D w D x The weighting is chosen to satisfy 1 11 12 5 (17) [(s)]  W (s) ,  y  C x  D w D x 1 2 21 22 5 (18) (21) where, The transfer function from disturbance to the state of the augmented system is 1 1   Tx z  sI [A  B2 K(s)C2 ] [B1  B2 K(s)D21]0 x  r 0 x  p   x  z  z y  y (22)  w  , p , p , where K(s) is controller. Three transfer  A 0  A  p functions (10)-(12) become B C A   w p11 w  H (s)  sE 0T  AC 2 xzr  Bp1   Bp2  H (s)  E 0T SD 1 xzr B1    B2    Bw Dp111 Bw Dp121   ,   , H (s)  E 0T  TD 3 xzr D C B  where C  w p11 w 1  D 0  E  1 0 0 0  w p12  , 1 , C  C 0 E2  0 1 0 0 E3  0 0 1 0 2  p2  , D D  D D  4. ADAPTIVE ROBUST CONTROL OF D  w p111 D  w p121 11  D  12  D  NONLINEAR PART  w p112 ,  w p122 In this part we will derive the controller for D  D D  D hydraulic actuator used in suspension system. 21 p21, 22 p22 The controller is designed based on adaptive robust control technique proposed by Bin Yao[3]. The H  control problem is to find an internal Consider hydraulic actuator dynamic equations stabilizing controller, K(s) , for the augmented (5)-(6). The parameter is considered as unknown system, G(s) , such that the inf-norm of the parameter  f  4e /Vt . The main reason for closed loop transfer function, Tzw , is below a choosing  f as unknown factor is that the bulk given positive scalar  modulus of hydraulic fluid is known to change dramatically even when there is a small leakage Find T   zw  between piston and cylinder. K(s)stabilizing (19) The equation (5) can be written in the form Furthermore, from the small gain theorem the x [a x  a (x  x )  robust stability of the closed loop system under 5 1 5 2 2 4 presence of parameter uncertainty is assured if a3 Ps A  sgn(x6 )x5 x6 ]  d   1. Here the change of the parameters of the (23) system is treated by multiplicative uncertainty Trang 8 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K1- 2015 2 a1  Ctm a  A 1  1  where ; 2 ;    a x  a (x  x )  (x  k z ) a a 1 5 2 2 4 ˆ 5d 1 1 3    a  C w A/  3 d f ; and  is unknown (28)  d 1 1 2 parameter; denotes disturbances and  r   z1   M (a1x5  a2 (x2  x4 )  4 min a3 11 their extents are known 2 1 2   a3 a )  d M    { :     } 12   min max (29) where | d | d M The adaptive control law can be obtained as k the following steps. 1 : tunable parameter Step 1: Let’s define ˆ   (ˆ)  b  a3 Ps Asgn(x6 )x5     (24) M max min  Equation (23) becomes ˆ  is estimated by  using the following x5 [a1x5  a2 (x2  x4 )  bx6 ] d adaptation law (25) ˆ   z [a x  a (x  x )  a  ] Define the error variable: 1 1 1 5 2 2 4 3 a , z  x  xd   0 1 5 5 (26) 1 (30)  To find a virtual control law  for x6 such  is a known arbitrary small positive number d that x5 tracks its desired value x5 using the and11,12are adjustable small positive numbers. procedure suggested in [3]. The term b , representing the nonlinear static gain between the Step 2: To find an actual control law for u flow rate and the valve opening x , is a function 6 x such that 6 tracks the desired control function  of x6 and also is non-smooth since x6 appears synthesized in step 1 with a guaranteed transient through a discontinuous function sgn(x6 ) . So a smooth modification is needed[3]. performance. Define the smooth projection  (ˆ) : Define the error variable    1     1 exp  (ˆ  ) (ˆ   ) z  x   max    max  max 2 6 (31)      ˆ  ˆ ˆ Adaptive robust control law consists of two  ()   ( [ min , max ])    1  parts: an adaptive part and a robust control part    1 exp (ˆ  ) (ˆ   )  min    min  min      u  u  u a r (32) The control law  is given by The adaptive part and robust control part are    a r (27) calculated as follows  a     The adaptive part and the robust control u   k z  p    a b  2 2 e ˆ 1 2c  part  r are calculated as follows    (33) Trang 9 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 18, No.K1- 2015  Table 1. Numerical values for simulation ur   z2h2 4b Parameters Values Units (34) where ms 290 kg w1 1 b 59 kg p  ˆ z  bx  xˆ  mu e  w 1  6 x 5 c 2 5 (35) bs 1000 Ns/m  ˆ xˆ  [a x  a (x  x )  bx ] 16812 N/m 5  1 5 2 2 4 6 ks (36) k 190000 N/m    t c  xˆ5  x t 5 5 (37)  f 4.515e13 N/m    w z  2c 1c 2 2 (38)  1.00   w z a x  a (x  x )  a   1.545e9 N/(m5/2kg1/2) 1c 1 1 1 5 2 2 4 3 2 (39) A 3.35e-4 m   w1 b  2   z1    a1x5  a2 (x2  x4 )  a3 Ps 10342500 N/m w2 x5 x5  (40) 1 Frequency domain h   2  2 2  M The plot of uncertainties and weighting 2 (41) functions are given in Fig. 3. Figures (4)-(6) show the gain plots for three transfer functions k2 , w1 , w2 and  2 are arbitrary positive (10)-(12) in cases of passive system, active system with desired force and actual force input. numbers. As shown in the figures, the designed nonlinear 5. SIMULATION RESULTS ARC controller can treat the nonlinearity and keep the H  frequency performance well. The numerical values using in this simulation are given in the Table 1[9]. 20 The weighting function is chosen as W ( j) 0 ( j) for bs  3.135s  9.2625  W (s) 0 -20  1   0  W (s)   0.93s  29 ( j) for      -40 ms  0 W  4  0 3.510  -60 Gain(dB) -80 ( j) for ks The controller is calculated with the value of -100 ( j) for    0.99 . The road velocity disturbance is -120 kt assumed to be from road displacement -140 10-2 10-1 100 101 102 r  0.1sin 2 f t . The parameters of ARC Frequency (Hz) controller are chosen to be  1  5e6 , k1  150 Fig. 3. Plots of uncertainties and weighting function , k2 10 ,   0.001, 11  5 , 12  2 ,  2  5 and d M  2 . Trang 10 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K1- 2015 30 6 20 4 10 2 0 0 -10 passive system Gain(dB) d -2 activesystemwith x5 input -20 * * * * * * activesystemwith x5 input -4 passive system -30 -6 Suspensiondeflection (mm) active system -40 -1 0 1 2 10 10 10 10 -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 -40 -50 2 -60 passive system 0 -70 Gain(dB) activesystemwith xd input -80 5 -2 -90 * * * * * * activesystemwith x5 input -4 passive system -100 deflection Tire (mm) -6 active system -110 -1 0 1 2 10 10 10 10 -8 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2 Frequency (Hz) Time (s) Fig. 5. Gain plots for suspension deflection transfer function Fig. 9. Tire deflection with step disturbance -25 1.5 -30 1 -35 ) 2 0.5 -40 0 -45 passive system Gain(dB) -0.5 -50 d activesystemwith x5 input -1 Acceleration(m/s -55 * * * * * * activesystemwith x5 input -1.5 passive system active system -60 10-1 100 101 102 -2 0 0.5 1 1.5 2 2.5 3 3.5 4 Frequency (Hz) Time (s) Fig. 6. Gain plots for tire deflection transfer function Fig. 10. Acceleration with sine disturbance 2 6 1.5 5 passive system 1 4 ) 2 active system 3 0.5 2 0 1 -0.5 0 -1 -1 Acceleration(m/s -2 deflectionSuspension (mm) -1.5 passive system active system -3 -2 0 0.5 1 1.5 2 2.5 3 3.5 4 -4 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2 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 2 system in case of sine wave disturbance are given 1.5 in Figs. (10)-(12). The road amplitude is assumed 1 to be 0.1 m with frequency of 1 Hz . At this 0.5 frequency, active system reduces considerably 0 the effects of disturbance. -0.5 6. CONCLUSION -1 Tire deflection Tire (mm) -1.5 This paper presents a control of active suspension passive system active system -2 system using and nonlinear adaptive robust 0 0.5 1 1.5 2 2.5 3 3.5 4 H  Time (s) control method. controller achieved the Fig. 12. Tire deflection with sine disturbance robustness with the presence of parameter uncertainties and minimized the effects of Time domain disturbance. The nonlinear ARC controller treats The responses of the system with step and well the non-linearity and the parameter sine wave disturbances are considered. uncertainties of hydraulic actuator. Simulation Responses of the system in case of step results show that the designed controller can keep disturbance are given in Figs. (7)-(9). The step the good performance of controller in both road velocity is of 0.1 m/s. Body acceleration and frequency and time domains. tire deflection are much reduced but the suspension deflection is higher. Responses of the Điều khiển hệ thống treo chủ động của xe ô tô dùng 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 khiển thích nghi bền vững (ARC) được của xe ô tô là một đề tài thú vị trong lĩnh sử dụng để thiết kế bộ điều khiển lực vực nghiên cứu về ô tô. Bài báo này đề bền vững với các thông số không biết xuất phương pháp điều khiển hệ thống chắc của bộ chấp hành. Kết quả mô treo chủ động bằng lý thuyết và điều phỏng đã thể hiện tính hiệu quả của bộ khiển thích nghi bền vững. Kỹ thuật điều 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]. T.T. Nguyen, V.G. Nguyen and S.B. Kim, [6]. T.T. 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Control Conference, pp. 1702-1706, San Diego, California USA, June 1999. Trang 13