Functional Based Adaptive and Fuzzy Sliding Controller for Non-Autonomous Active Suspension System

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(1)1027. Functional Based Adaptive and Fuzzy Sliding Controller for Non-Autonomous Active Suspension System∗ Shiuh-Jer HUANG∗∗ and Hung-Yi CHEN∗∗∗ In this paper, an adaptive sliding controller is developed for controlling a vehicle active suspension system. The functional approximation technique is employed to substitute the unknown non-autonomous functions of the suspension system and release the model-based requirement of sliding mode control algorithm. In order to improve the control performance and reduce the implementation problem, a fuzzy strategy with online learning ability is added to compensate the functional approximation error. The update laws of the functional approximation coefficients and the fuzzy tuning parameters are derived from the Lyapunov theorem to guarantee the system stability. The proposed controller is implemented on a quarter-car hydraulic actuating active suspension system test-rig. The experimental results show that the proposed controller suppresses the oscillation amplitude of the suspension system effectively.. Key Words: Active Suspension System, Functional Approximation, Fuzzy Compensation. 1.. Introduction. The objective of an active suspension system is to provide the ride comfort and handling capability under different road disturbances. In recent years, various control methods have been developed to improve the performance of the active suspension system. Alleyne and Hedrick(1) designed an adaptive controller with a modified adapting scheme to reduce the model error and to cope with the system uncertainties. Chantranuwathana and Peng(2) proposed an adaptive robust force controller to tackle the actuator uncertainties of active suspension systems. Yoshimura et al.(3) designed an sliding mode controller for active suspension system where the sliding surface was derived from linear quadratic control theory. Fialho and Balas(4) combined the linear parameter-varying control and nonlinear backstepping technique to design a road adaptive active suspension. Smith and Wang(5) derived a parameterized stable controller for a vehicle active suspension system with fixed prespecified closed-loop transfer function. Sam et al.(6) proposed a proportionalintegral sliding mode control scheme to control a quarter∗ ∗∗. ∗∗∗. Received 27th January, 2006 (No. 05-5149) Department of Vehicle Engineering, National Taipei University of Technology, No.1, Sec.3, Chung-Hsiao E. Rd. Taipei 106, Taiwan. E-mail: [email protected] Department of Mechanical Engineering, Mingchi University of Technology, No.84, Gungjuan Road, Taishan, Taipei 243, Taiwan. JSME International Journal. car active suspension system. Since it is difficult to establish an accurate dynamic model of a physically active suspension system for designing a model-based controller, various model-free approaches have been developed in this application field. Cherry and Jones(7) employed a fuzzy logic technique to control an automotive suspension system with continuously varying damping. D’Amato and Viassolo(8) developed the fuzzy controller to enhance the riding comfort for an active suspension system. Huang and Chao(9) designed a grey fuzzy controller, whereas Nizar et al.(10) developed a sliding mode neural network inference fuzzy controller for a vehicle active suspension system. Huang and Lin proposed a self-organizing fuzzy controller(11) and an adaptive fuzzy sliding mode controller(12) to control the oscillation amplitudes of sprung mass position and acceleration with the learning ability to tackle the system dynamics and road conditions variations. Chen and Huang(13) proposed a functional approximation based sliding controller for controlling a non-autonomous quartercar suspension system with time-varying loadings. The functional approximation technique is employed to model the system unknown functions for releasing the modelbased limitation of sliding mode control strategy. The approximation error depends on the number of Fourier series basis functions and the weighting vectors. If we want to improve the functional approximation accuracy, the number of Fourier series functions will increase significantly. That will increase the computation burden and Series C, Vol. 49, No. 4, 2006.

(2) 1028 system complicacy, and hinder the practical implementation. It is well known that the approximation error can be reduced by introducing a compensator into the system. Hence, a functional approximation based adaptive sliding controller with a fuzzy compensator is proposed to control a quarter-car hydraulic actuating suspension system. 2. Active Suspension System Model and Dynamics A quarter-car hydraulic actuating active suspension system shown in Fig. 1 is built for investigating the dynamic behavior and control performance. Its dynamic characteristic can be simplified into a 2-DOF dynamic model as Fig. 2. The car body is considered as a 210 kg sprung mass m s . The passive components of this suspension system consist of a damper b s = 2 900 N(s/m) and a spring k s = 4 000 N/m. The function of the tire is simulated by a spring with coefficient kt and an unsprung mass mu . The actuating force Fa is generated by the hydraulic actuator. It can provide 40 000 N actuating force. The variables Z s and Zu represent the vertical displacements of sprung mass and unsprung mass, respectively. Zr is the vertical profile of road surface. If the tire is always contacted with the road surface and the suspension travel is below its physical limit, the dynamic equations of this quarter-car active suspension system can be described as. Fig. 1 Hydraulic actuating quarter-car active suspension system. Fig. 2. Quarter-car active suspension system model. Series C, Vol. 49, No. 4, 2006. Fa − F f = m s Z¨ s + b s (Z˙ s − Z˙u ) + k s (Z s − Zu ) (1) F f − Fa = mu Zü + b s (Z˙u − Z˙ s ) + k s (Zu − Z s ) + kt (Zu − Zr ) (2) where Fa and F f are the hydraulic actuating force and the hydraulic friction force, respectively. The time derivative of the hydraulic actuating force can be represented as(12) F˙ a (t) = P˙ L (t)AP = AP (4B/Vt )[Kg (t)Kv u(t) −CT PL (t) − AP (Z˙ s (t) − Z˙u (t))]. (3). where AP is the cross section area of hydraulic cylinder, PL (t) is the cylinder differential pressure, Kv is the servovalve gain, Kg is the servovalve flow gain, Vt is the total compressed volume, B is the bulk modulus of the hydraulic oil, CT is the total leakage coefficient and u(t) is the servovalve control voltage. To simplify this model description, the non-autonomous system dynamics can be represented as the following third order model. ... x 1 (t) = f (X,t) + b(t)u(t) + d(t) (4) where x1 = Z s is the sprung mass displacement, X is the state vector, f (X,t) is a function of state variables, b(t) is the control gain, d(t) is the system uncertainty and disturbance, and u(t) is the servovalve control voltage. The function f (X,t) is an unknown time-varying function whose variation bound is unknown and b(t) is an unknown control gain function. Since the system dynamics has nonlinear behavior and contains time-varying uncertainties with unknown bounds, the functional approximation technique is employed to approximate these unknown functions f (X,t) and b(t) for releasing the model requirement. If the system uncertainties are piecewise continuous and satisfy the Dirichlet conditions, the functional approximation technique can be used to represent these system uncertainties with the combination of finite number of weighted Fourier series basis functions. The theoretical background and evaluation of function approximation were developed by Ge et al.(16) In addition, the adaptive fuzzy sliding control strategy is introduced to compensate the approximation error for improving the control performance. 3. Controller Design and Stability Analysis The hydraulic actuating suspension system has nonlinear time-varying dynamics. It is difficult to estimate an accurate dynamic model for model based controller design. Here, the functional approximation technique is employed to simulate the unknown nonlinear functions for sliding mode control design. In addition, an adaptive fuzzy compensation strategy is introduced to tackle the approximation error and system dynamics uncertainty. The system control block diagram of the quarter-car active suspension system is shown in Fig. 3. The adaptive laws of the function coefficients and the adaptive rules of fuzzy sliding controller can be derived from Lyapunov stability JSME International Journal.

(3) 1029. Fig. 3 Control block diagram. theorem. The sliding surface of this third order suspension system is defined as 2 d s= + λ x1 = x¨1 + 2λ x˙1 + λ2 x1 (5) dt where the positive parameter λ implies the convergent rate of x1 on the sliding surface. The time derivative of s can be derived as ... s˙ = x 1 + 2λ x¨1 + λ2 x˙1 (6). (a). Substituting Eq. (4) into (6) yields s˙ = f (X,t) + b(t)u + 2λ x¨1 + λ2 x˙1 = b[ f b−1 + u + b−1 (2λ x¨1 + λ2 x˙1 )]. (7). In order to obtain the sliding surface reaching condition and achieve the approximation error compensation, the control law u(t) can be designed as u(t) = uFA (t) + uAFSMC (t) = − fâ − bˆ −1 (2λ x¨1 − λ2 x˙1 ) −C T φ. (8). where fâ is the estimate value of fa = f b−1 and bˆ is the estimate of b. In this development, both fâ and bˆ have to be estimated on-line. If bˆ is close to zero, the control law will grow unbounded. Therefore, a lower bound of b should be specified as b ≥ b > 0. Then, an adaptive fuzzy sliding-mode control (AFSMC) strategy(12) is employed to compensate the functional approximation error and system variation. The tracking error e(t) and the change rate of error e˙ (t) are combined together to define a sliding surface variable sAFSMC for the only input of the fuzzy compensator. The term C T φ in Eq. (8) is the adaptive fuzzy compensation component which can be calculated from the fuzzy inference decision and defuzzification operation. Triangular membership functions shown in Fig. 4 (a) is used to classify the fuzzy input and output variables. The one-dimension fuzzy rules of AFSMC are JSME International Journal. (b) Fig. 4 (a) Membership functions of the errors and error changes (b) Fuzzy rules of AFSMC. shown in Fig. 4 (b). A scaling factor gs is employed to map the sliding surface variable into this fuzzy universe of discourse. Substituting Eq. (8) into (7), obtain s˙ = b[( fa − fâ ) + (b−1 − bˆ −1 )(2λ x¨1 + λ2 x˙1 ) −C T φ] (9) where fa , fâ , b−1 and bˆ −1 are assumed to be unknown bounded piece-wise continuous functions and satisfy the Dirichlet conditions. Then, they can be represented by the functional approximation technique(16) as fa = W Tfa Z fa. (10). ˆ Tf Z fa fâ = W a. (11). b−1 = WbTa Zba. (12). ˆ −1. b. ˆ bT Zba =W a. (13). ˆ fa , Wba , W ˆ ba ∈ are weighting vectors and where W fa , W Z fa , Zba ∈ n are the vectors of basis Fourier function. Hence, the Eq. (9) can be rewritten as n. Series C, Vol. 49, No. 4, 2006.

(4) 1030 ˜ Tf Z fa + W ˜ bT Zba (2λ x¨1 + λ2 x˙1 ) −C T φ] s˙ = b[W a a where ˆ Tf ˜ Tf = W Tf − W W a a a ˜ bT W a. = WbTa. (14) (15). ˆ bT −W a. (16). To prove the stability of this control system and find ˆ ba and C, a Lyapunov ˆ fa , W the update laws for vectors W function candidate is chosen as ˜ ba ,C) ˜ fa , W V(s, W 1 ˜T b T 1 ˜T ˜ ˜ C C = s2 + b[W fa Q fa W fa + Wba Qba Wba ] + 2 2 2γ (17) where Q fa , Qba ∈ n×n are symmetric positive definite matrices. By taking the time derivative of the Lyapunov function candidate, we can obtain ˜ ba ,C) ˙ W ˜ fa , W V(s, ˜ bT Qba W ˜ Tf Q fa W ˜˙ fa + W ˜˙ ba ] + b C T C˙ (18) = s s˙ + b[W a a γ T T T T ˙ˆ and W ˙ˆ , the Eq. (18) can ˙˜ = −W ˙˜ = −W Since W fa. fa. ba. ba. be rewritten as. ˙ˆ ) ˜ ba ,C) = b W ˜ Tf (Z fa s − Q fa W ˙ W ˜ fa , W V(s, fa a ˙ˆ ] −C T (sφ − 1 C) ˜ bT [Zba s(2λ x¨1 +λ2 x˙1 ) − Qba W ˙ +W ba a γ (19) ˆ ˆ The update laws for W fa , Wba and C are chosen as. ˙ˆ = Q−1 Z s W (20) fa fa fa  −1  Qba Zba s(2λ x¨1 + λ2 x˙1 ) if 0 < bˆ −1 < b−1       2  ˙1 )  Q−1 ba Zba s(2λ x¨1 + λ x ˆ˙ ba =  W (21)   −1 −1  ˆ  if b ≥ b & s(2λ x¨1 + λ2 x˙1 ) < 0       0 if bˆ −1 ≥ b−1 & s(2λ x¨1 + λ2 x˙1 ) ≥ 0 C˙ = γsφ − k |s|C (22) where γ is the positive learning rate and k is a positive parameter for the introducing e-modification term of update law(14) . Then, the Eq. (19) can be further rewritten as  k    −b |s|C T C ≤ 0   γ      if 0 < bˆ −1 < b−1       k    −b |s|C T C ≤ 0    γ      if bˆ −1 ≥ b−1 &     ˙ W ˜ fa , W ˜ ba ,C) =  V(s, s(2λ x¨1 + λ2 x˙1 ) < 0 (23)        k   ˜ bT Zba ) −b |s|C T C + b(W   a  γ     2  × s(2λ x¨1 + λ x˙1 ) ≤ 0        if bˆ −1 ≥ b−1 &       s(2λ x¨ + λ2 x˙ ) ≥ 0 1. where Series C, Vol. 49, No. 4, 2006. 1. ˜ bT Zba = b−1 − bˆ −1 < 0 W a. (24). Hence, the convergence of the system output error with the control law u(t) of Eq. (8) can be guaranteed and proven by using Barbarlet’s lemma(15) . 4. Experimental Results In order to investigate the control performance of the proposed controller, the following experiments are performed. The sampling frequency is 100 Hz. The parameter λ of the sliding surface is chosen as 1. The Lyapunov function weighting matrices are chosen as constants Q fa = 3.33 × 10−3 [I] and Qba = 100 [I], respectively, to adjust the converging speed of functional approximation parameters and achieve good control performance. The first five terms of the Fourier series functions are chosen as the functional approximation basis functions. The AFSMC’s sliding surface variable was divided into eleven equal-span fuzzy input subsets within [−1, +1] with triangular membership function as Fig. 4 (a). A parameter gs = 2 was chosen to map the fuzzy input variable into that normalized range based on experimental test. The strictly positive parameter ρ in Fig. 3 for constituting the sliding surface variable of fuzzy compensator was selected as 1.2 to set the weightings of position and velocity errors in the sliding variable. The learning rate parameter γ was set as 5. It can be designed to adjust the AFSMC variation situation. The central positions of the fuzzy output membership functions were all initialized at zero. When a vehicle is riding on a concave-convex terrain with 40 mm height as the dotted line shown in Fig. 5, the dynamic response of the sprung mass position by using functional approximation (F.A.) based sliding mode controller and the proposed F.A.+AFSMC control strategy are shown in Fig. 6 for comparison. The dotted line exhibits the sprung mass response by using the F.A. based control and the solid line depicts the position response of the proposed controller with additional AFSMC compensa-. Fig. 5. Sprung mass displacement (concave-convex terrain) (solid line: F.A.+AFSMC; dotted line: Road profile) JSME International Journal.

(5) 1031. Fig. 6 Sprung mass displacement (concave-convex terrain) (solid line: F.A.+AFSMC; dotted line: F.A. only). Fig. 8 Sprung mass displacement (random terrain) (solid line: F.A.+AFSMC; dotted line: Road profile). Fig. 7 PSD of sprung mass acceleration (concave-convex terrain) (solid line: F.A.+AFSMC; dashed line: F.A. only). Fig. 9 Sprung mass displacement (random terrain) (solid line: F.A.+AFSMC; dotted line: F.A. only). tion. It can be observed that the maximum sprung mass displacement has been reduced from 1.1 mm to 0.5 mm by introducing the AFSMC compensator. The root mean square (RMS) values of the sprung mass displacement are 0.36 mm and 0.15 mm for the F.A. based controller only and the proposed F.A. based control with AFSMC compensation, respectively. Figure 7 shows the power spectra density (PSD) of the vertical acceleration of the sprung mass. The dotted line exhibits the sprung mass acceleration by using the F.A. based controller only, and the solid line depicts the acceleration of the proposed controller with additional AFSMC compensation. It can be observed that the control performance has been improved within the frequencies between 0.04 Hz and 10 Hz by adding the AFSMC compensation loop into the functional approximation technique. When a vehicle is riding on a random terrain as the dotted line shown in Fig. 8, the dynamic response of the sprung mass vertical position by using F.A. based sliding mode controller and the proposed F.A.+AFSMC control. strategy are shown in Fig. 9 for comparison. The dotted line exhibits the sprung mass displacement by using the F.A. based control and the solid line depicts the displacement of the proposed controller with additional AFSMC compensation. It can be observed that the maximum sprung mass displacement has been reduced from 1.05 mm to 0.65 mm by introducing the AFSMC compensator. The root mean square (RMS) values of the sprung mass displacement are 0.33 mm and 0.18 mm for the F.A. based controller only and the proposed functional approximation based control with AFSMC compensation, respectively. Figure 10 shows the power spectra density (PSD) of the vertical acceleration of the sprung mass. The dotted line exhibits the sprung mass acceleration by using the F.A. based controller only, and the solid line depicts the acceleration of the proposed controller with additional AFSMC compensation. It can be observed that the control performance has been improved within the frequencies between 0.04 Hz and 10 Hz by adding the AFSMC compensation loop into the functional approximation technique.. JSME International Journal. Series C, Vol. 49, No. 4, 2006.

(6) 1032. (3). (4). (5). Fig. 10 PSD of sprung mass acceleration (random terrain) (solid line: F.A.+AFSMC; dotted line: F.A. only). 5. Conclusion A functional approximation based adaptive sliding controller with fuzzy compensation has been successfully employed to control a quarter-car hydraulic actuating active suspension system. From the experiment results, it can be observed that the proposed control scheme has significantly suppressed the position and acceleration oscillation amplitudes of the sprung mass for improving the ride comfort and handling capability. The introduction of a fuzzy compensator into the proposed F.A. based adaptive sliding controller with small number of Fourier series functions can improve the system control performance and compensate the approximation error significantly. Hence, this model-free adaptive sliding controller can be employed in nonlinear systems with unknown information by using a simple functional approximation control structure and a fuzzy compensation loop. It can significantly reduce the computation and database burden. This approach can release the model-based requirement of sliding mode control and simplify the practical implementation problem of the model-free functional approximation sliding mode control.. (6). (7). (8). (9). (10). (11). (12). (13). Acknowledgements This research was supported by the National Science Council of the Republic of China, with contract NSC932212-E-011-003.. (14). References. (15). (1). (2). Alleyne, A. and Hedrick, J.K., Nonlinear Adaptive Control of Active Suspensions, IEEE Transactions on Control Systems Technology, Vol.3, No.1 (1995), pp.94–101. Chantranuwathana, S. and Peng, H., Adaptive Robust. Series C, Vol. 49, No. 4, 2006. (16). Control for Active Suspensions, Proceedings American Control Conference, (1999), pp.1702–1706. Yoshimura, T., Kume, A., Kurimoto, M. and Hino, J., Construction of an Active Suspension System of a Quarter Car Model Using the Concept of Sliding Mode Control, Journal of Sound and Vibration, Vol.239, No.2 (2001), pp.187–199. Fialho, I. and Balas, G.J., Road Adaptive Active Suspension Design Using Linear Parameter-Varying GainScheduling, IEEE Transactions on Control Systems Technology, Vol.10, No.1 (2002), pp.43–54. Smith, M.C. and Wang, F.-C., Controller Parameterization for Disturbance Response Decoupling: Application to Vehicle Active Suspension Control, IEEE Transactions on Control Systems Technology, Vol.10, No.3 (2002), pp.393–407. Sam, Y. Md., Osman, J.H.S. and Ghani, M.R.A., A Class of Proportional-Integral Sliding Mode Control with Application to Active Suspension System, System and Control Letters, Vol.51 (2004), pp.217–223. Cherry, A.S. and Jones, R.P., Fuzzy Logic Control of an Automotive Suspension Systems, IEE Proceedings D: Control Theory Application, Vol.142, No.2 (1995), pp.149–160. D’Amato, F.J. and Viassolo, D.E., Fuzzy Control for Active Suspensions, Mechatronics, Vol.10 (2000), pp.897–920. Huang, S.J. and Chao, H.C., Fuzzy Logic Controller for a Vehicle Active Suspension System, Proceedings of the Institution of Mechanical Engineers, D, Vol.214 (2000), pp.1–12. Nizar, A.-H., Tarek, L., Dae, S.J., Jonathan, W. and Faysal, A.-A., Sliding Mode Neural Network Inference Fuzzy Logic Control for Active Suspension Systems, IEEE Transactions on Fuzzy Systems, Vol.10, No.2 (2002), pp.234–246. Huang, S.J. and Lin, W.C., A Self-Organizing Fuzzy Controller for an Active Suspension System, Journal of Vibration and Control, Vol.9 (2003), pp.1023–1040. Huang, S.J. and Lin, W.C., Adaptive Fuzzy Controller with Sliding Surface for Vehicle Suspension Control, IEEE Trans. on Fuzzy Systems, Vol.11, No.4 (2003), pp.550–559. Chen, P.-C. and Huang, A.-C., Adaptive Sliding Control of Non-Autonomous Active Suspension Systems with Time-Varying Loadings, Journal of Sound and Vibration, Vol.282 (2005), pp.1119–1135. Narendra, K.S. and Annaswamy, A.M., A New Adaptive Law for Robust Adaptation without Persistent Excitation, IEEE Trans. on Automatic Control, Vol.AC32, No.2 (1987), pp.134–145. Narendra, K.S. and Annaswamy, A.M., Stable Adaptive Systems, (1989), Prentice Hall, Englewood Cliffs, NJ. Ge, S.S., Hang, C.C., Lee, T.H. and Zhang, T., Stable Adaptive Neural Network Control, (2002), Kluwer Academic Publishers.. JSME International Journal.

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