Fuzzy control for active suspensions

Mechatronics 10 (2000) 897±920 Fuzzy control for active suspensions Fernando J. D'Amato*, Daniel E. Viassolo School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, 47907-1282, USA Received 27 May 1998; received in revised form 26 August 1999; accepted 13 September 1999 Abstract A methodology for the design of active car suspension systems is presented. The goal is to minimize vertical car body acceleration, for passenger comfort, and to avoid hitting suspension limits, for component lifetime preservation. A controller consisting of two control loops is proposed to attain this goal. The inner loop controls a nonlinear hydraulic actuator to achieve tracking of a desired actuation force. The outer loop implements a fuzzy logic controller which interpolates linear locally optimal controllers to provide the desired actuation force. Final controller parameters are computed via genetic algorithmbased optimization. A numerical example illustrates the design methodology. 7 2000 Elsevier Science Ltd. All rights reserved. 1. Introduction The basic idea in active control of suspensions is to use an active element (the actuator, e.g., a hydraulic cylinder) to apply a desired force between the car body and the wheel axle. This desired force is computed by the car's control unit to achieve certain performance objectives under external disturbances (e.g., passenger comfort under road imperfections). Although passive suspension systems can e€ectively handle some control objectives, active suspension systems are currently replacing (or being combined with) them. This happens mainly because active systems o€er more design ¯exibility, and thus the range of achievable objectives increases. * Corresponding author. Fax: +1-765-494-0307. E-mail address: fdamato@ecn.purdue.edu (F.J. D'Amato). 0957-4158/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 7 - 4 1 5 8 ( 9 9 ) 0 0 0 7 9 - 3 898 F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 Active suspension systems reduce car body accelerations by allowing the suspension to `absorb' wheel accelerations. This is achieved at the expense of using more suspension travel, and thus increasing the probability of hitting the suspension rattle-space limits. Hitting the rattle-space limits translates into both passenger discomfort and wear of vehicle components [1]. These considerations motivate us to investigate the ride/rattle-space trade-o€ for active suspensions, which seems to have been overlooked until the work in [1]. This paper focuses on the (loosely stated) problem: Design an active suspension system to avoid hitting the rattle-space limits for `large' suspension travels, and to minimize the car body acceleration for `small' suspension travels. We present an approach for solving the problem stated above. Our approach uses both an inner control loop and an outer control loop. The inner loop controls the hydraulic actuator to track a desired actuation force. The outer loop implements a fuzzy logic controller, with parameters computed by a genetic algorithm (GA)-based optimization. Our major contribution is a detailed design methodology to solve this particular class of problems. The approach combines di€erent control design techniques to achieve performance dependent of the perturbation size. The methodology is applied to a quarter-car suspension system. The remaining part of this paper is organized as follows. Section 2 contains a description of the model for the suspension system. The control problem is set up in Section 3. In Section 4 we present a design methodology for active suspension systems, and the speci®c details on each step are given in Section 5. The resulting controller designs are given in Section 6, and the conclusions in Section 7. 2. Mathematical model The mathematical model for the suspension dynamics is taken from [2,3] and is brie¯y described as follows. Consider the quarter-car passive/active suspension system shown in Fig. 1. The wheel and the (massless) axle are connected to the car body through the spring±damper±actuator combination. The tire is modeled as a simple spring. The dynamics for the suspension system, excluding the actuator and when the suspension travel is below its physical limit, is described by the linear equation 3 3 2 2 3 2 0 1 0 0 0 0 2 3 6 72 z 3 6 ÿKt 7 6 1 7 Kt z_1 7 1 7 6 ÿKa ÿCa 6 7 6 0 7 7 6 7 6 6 7 6 z_2 7 6 M M M M z M w w 7 7 6 6 7 6 2 7‡6 6 7ˆ6 1† 76 7F 7r ‡ 6 4 5 4 z_3 5 6 0 7 z3 60 7 60 7 0 0 1 7 7 6 6 7 6 z_4 5 z4 4 Ka 4 Kt 5 4 ÿ1 5 Ca ÿKt 0 Mw Mw Mw Mw Mw F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 899 or, in compact form, z_l ˆ Azl ‡ B1 r ‡ B2 F, where z1 : ˆ x b ÿ x w is the suspension travel (body displacement xb relative to wheel displacement xw), z2 : ˆ z_1 , z3 : ˆ x w , and z4 : ˆ z_3 : The inputs are the road disturbance (displacement) r and the actuator force F. The suspension travel is physically constrained according to j z1 t† j Rz1 , 8tr0: 2† The parameters of this model are: the body mass Mb, the wheel mass Mw, the damping coecient Ca, the spring coecient Ka, the tire spring coecient Kt and the hard limit for the suspension travel z1 ; the parameter M is de®ned by 1=M ˆ 1=Mw ‡ 1=Mb : The dynamics of the hydraulic actuator is given by F_ ˆ ÿbF ÿ aL 2 z2 ‡ gLN F, u†u, 3† where r  F F j Ps ÿ sgn u† j, N F, u†: ˆ sgn Ps ÿ sgn u† L L L is the piston area, Ps is the supply pressure, a, b, g are constant coecients, and Fig. 1. Quarter-car model with hydraulic actuator and spring/damper. 900 F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 u is the input voltage for the servo-valve. The servo-valve dynamics is considered fast enough so that it can be modeled as a constant. (For a more detailed description of the hydraulic actuator see [2,3].) Numerical values for the model parameters are taken from [2,3], and are given below M Mw Ka Ca Kt z1 a b g Ps L 49 kg 59 kg 16,812 N/m 1000 Ns/m 190,000 N/m 8 cm 4.515  1013 N/m5 1 sÿ1 5 1 1.545  109 N= m 2 kg 2 † 10,342,500 Pa 3.35  10ÿ4 m2 3. Control problem The control objective is to maximize the passenger comfort, while preserving the lifetime of suspension components, under road disturbances. The comfort is determined by the level of vertical car body acceleration xÈb experienced by the passenger. The lifetime of components is preserved by avoiding hitting the rattlespace limits (i.e., avoiding j z1 jˆ z1 ). Hence, the controller objectives can be stated as (a) increase, with respect to open-loop, the range of `typical' road disturbances for which the suspension travel limits are not reached, and (b) minimize the peaks of the body accelerations under road disturbances. Consider the system de®ned by Eqs. (1)±(3), and let the set of typical road disturbances be of the form  a 1 ÿ cos 8pt††=2 if 0:50RtR0:75 4† r t† ˆ 0 otherwise, where a denotes the `bump' amplitude (see Fig. 2). This is the type of disturbance considered, for example, in [1]. The control problem is precisely de®ned as follows: Design a control law u t† ˆ C z t††, with z: ˆ ‰z1 z2 z3 z4 F ŠT , for the system in Eqs. (1)±(3) such that the closed-loop response to inputs of the form (4), with z(0)=0, satis®es: the suspension travel vz1v does not reach the limit z-1 for bumps of amplitudes a R 11 cm, and F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 901 Fig. 2. Typical road disturbance Ð bump with amplitude 11 cm. the peak of the body acceleration max t j x b t† j is minimized for bumps of amplitudes a R 5 cm.1 4. Design methodology To solve the control problem stated in the previous section we propose the following two-step procedure: 1. design an inner loop controller for tracking actuator commands, and 2. design an outer loop controller to satisfy performance requirements. The inner loop controller produces the actuator input u which makes the actuator force F track the force command Fcmd. This controller attempts to cancel the actuator nonlinearities. The outer loop controller produces the force command Fcmd to achieve performance. This controller has performance requirements that 1 The numerical bounds of 11 cm and 5 cm for a are suggested from results in [1] and from the authors' preliminary designs [4]. 902 F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 depend on the size of the perturbations. Namely, we have one performance objective when the perturbations are relatively small, and another performance objective when the perturbations are large. For this outer loop controller we propose a fuzzy architecture, to schedule the control action according to the perturbation size. A representation of this con®guration is shown in Fig. 3 and the following section gives details on each step of the design methodology. 5. Methodology speci®cs 5.1. Inner loop controller Given a command signal Fcmd for the actuator force F, the proposed inner loop controller has the form shown in Fig. 3, and is described by uˆ n r satF1 F † gL Ps ÿ sgn n† L where n is de®ned by  z_c ˆ ÿlc zc ÿ kc lc F ÿ satF1 F †† n ˆ aL 2 z2 ‡ b ÿ lt †satF1 F † ‡ lt Fcmd ‡ zc † ‡ F_cmd ‡ z_c 5† 6† and satF1(
) is the standard saturation function with saturation level F1, and F1 is a force amplitude slightly less than PsL. Fig. 3. Schematic of interconnection. F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 903 The rationale for choosing this inner loop controller is given as follows. If the force F generated by the actuator never exceeds the value F1, the controller Eq. (5) constitutes a feedback linearization controller (see, for example [5,6]) that renders linear the input±output dynamics from the actuator command Fcmd to the measurements zl. More explicitly, replacing (5) in (3) when j F j< F1 we obtain the following dynamics for the tracking error e: ˆ F ÿ Fcmd : e_ ˆ ÿlt e, i:e:, F_ ˆ ÿlt F ÿ Fcmd † ‡ F_cmd : 7† Then, the u in Eq. (5) cancels the dynamics in Eq. (3) and assigns the tracking error dynamics, given by lt. On the other hand, if the actuator force magnitude exceeds the value F1, the state zc in (6) becomes excited and attempts to restore F back to the `linearization interval' [ÿF1,F1]. To obtain dynamics for the tracking error e and the controller state zc that are faster than the dynamics in (1), we choose lt ˆ 1260, lc ˆ 2000, kc ˆ 10: 8† Similar techniques for the design of tracking controllers for nonlinear hydraulic actuators can be found in [2,3]. 5.2. Outer loop controller The proposed outer loop controller is a so called parallel distributed compensator (PDC) based on a Takagi±Sugeno (TS) representation of the input/ output description of the dynamics from the inputs [r Fcmd]T to the performance output e (to be de®ned later). This TS representation can be interpreted as an interpolation of a set of linear systems (see [7]). The proposed PDC consists in an interpolation of optimal controllers designed for each of the rules of the TS model. In the next subsections, we describe each of the steps to design the outer loop controller. 5.2.1. Takagi±Sugeno fuzzy system With the inner loop controller described in Section 5.1 in place, the input/ output dynamics from r, Fcmd to zl (see Fig. 3) is approximately linear and equivalent to (1) with F replaced by Fcmd, i.e., z_l ˆ Azl ‡ B1 r ‡ B2 Fcmd : 9† In the following, the fuzzy Takagi±Sugeno (TS) description is used to accommodate the nonlinear (operating region-dependent) performance requirements for the linear dynamics (9). According to Section 3, for small disturbances the performance is measured in terms of body acceleration, while for large disturbances the performance is measured in terms of suspension travel. The suitability of the TS description is justi®ed as follows. FIRST, it allows the 904 F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 speci®cation of di€erent performance outputs2 for di€erent disturbance amplitudes. Second, it provides the structure to schedule a set of optimal controllers for the local performance objectives. A key observation is that the scheduling in terms of the perturbation size a in (4) can be reformulated in terms of the suspension travel amplitude vz1v. Our TS description is based on the linguistic variable (see [8]) suspension travel amplitude h, de®ned by the set 1 m fh, U, fT 1h , . . . , T m h g, fmh , . . . , mh gg where U ˆ ‰0, z1 Š is the universe of discourse of h, T ih are the term sets of h, and mih are the corresponding membership functions (to be discussed shortly). In what follows, we will denote with Xi the region of the state-space zl associated with the term set T ih of the linguistic variable de®ned above. For each region Xi we choose a corresponding linear nondimensional local performance output e i following the guidelines: If h is small, then e i has large contribution of xÈb and small contribution of vz1v. If h is large, then e i has small contribution of xÈb and large contribution of vz1v. This local performance output is de®ned by 2 ei 6 e1i i e ˆ4 2 eiFcmd 2 x b 3 6 li 10 m=s 2 6 6 z1 7 6 5 ˆ 6 1 ÿ li † 6 8 cm 6 4 Fcmd 13,000 N 3 7 7 7 7 i ˆ 1, . . . , m 7, 7 li 2 ‰0, 1Š, l1 ˆ 1, lm ˆ 0, 7 5 10† where li provides relative weight between suspension travel and body acceleration. Clearly, for the region of small suspension travel li 1 1, while for the region of large suspension travel li 1 0. The m rules of the TS description de®ning the linear dynamics and the local performance output e i have the form: Rule i: IF h is T ih , THEN z_l ˆ Azl ‡ B1 r ‡ B2 Fcmd ei ˆ Ci zi ‡ Di Fcmd , 11† 2 The vector of performance outputs contains all the signals we want to be `small', e.g., the body acceleration. F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 905 for i = 1, . . .,m, where A, B1 and B2 are de®ned in (1), and Ci and Di are de®ned by Eqs. (1) and (10). 5.2.1.1. Membership functions. The individual membership functions considered are piecewise combinations of 7th order polynomials. Each membership function is parameterized in terms of the transition abscissas, as described next. The membership function for the ®rst term set T 1h is given by 8 if 0RhRc1 <1 12† m1h h† ˆ 1 ÿ pc1 , d1 h† if c1 RhRd1 : 0 if d1 Rh, where pq, s h† is a 7th order polynomial parameterized by q and s such that pq, s q† ˆ 0 pq,0 s q† ˆ pq,00 s q† ˆ pq,000 s q† ˆ 0 pq, s s† ˆ 1 pq,0 s s† ˆ pq,00 s s† ˆ pq,000 s s† ˆ 0 For the term sets T ih , i ˆ 2, 8 0 > > > > < pai , bi h† mih h† ˆ 1 > > > 1 ÿ pci , di h† > : 0 . . ., m ÿ 1†, mih is given by if if if if if 0RhRai ai RhRbi bi RhRci ci RhRdi di RhRz1 , 13† m and for the last term set T m h , mh is given by 8 if 0RhRam <0 p h† if am RhRbm h† ˆ mm h : am , bm 1 if br RhRz1 : 14† We impose the following constraints for the transition abscissas ai ˆ ciÿ1 , bi ˆ diÿ1 , for i ˆ 2, 3, . . . , m: A sketch of these membership functions is shown in Fig. 4. The set of membership functions for the TS model is parameterized by the abscissas ai , bi , ci , di , which are collected in the membership function parameter vector pm : ˆ ‰a2


am b2


bm c1


cmÿ1 d1


dmÿ1 Š: 15† 5.2.1.2. TS representation. Finally, the TS representation that accommodates the nonlinear performance requirements is given by the linear dynamics (9) and the following `fuzzy blending' of the performance output: 906 F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 e t† ˆ C t†zl t† ‡ D t†Fcmd t†, 16† with C t†: ˆ m X Zi t†Ci iˆ1 D t†: ˆ m X Zi t†Di , 17† iˆ1 where Zi t†: ˆ mih m X h t†† j mh , i ˆ 1, 2, . . . , m: 18† h t†† jˆ1 (Notice, from Eq. (18) we have Zi t†r0, 8i, and Sm iˆ1 Zi t† ˆ 1, 8tr0:) Fig. 4. Membership functions m 1, m 2, m 3 in a hypothetical universe of discourse U=[0,15]. Vertical lines indicates the transition abscissas. F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 907 The performance output e is parameterized in terms of li and ri, for i = 1,2, . . . ,m, which are collected in the performance parameter vector pp : ˆ ‰l1


lm r1


rm Š: 19† In this manner we obtain a TS representation parameterized both by pm in Eq. (15) and by pp in Eq. (19). 5.2.2. Parallel distributed compensator To control the TS model introduced in Section 5.2.1 we use a parallel distributed compensator (PDC) [9,10]. This PDC is obtained by (a) designing a static state-feedback controller Ki for each rule i = 1,2, . . . ,m, (as detailed later) and by (b) fuzzy blending these Kis. A key point is that for the blending we utilize the same fuzzy sets as for the TS model. To be more precise, the actuator force command becomes ! m X Zi t†Ki zl t† 20† Fcmd t† ˆ K t†zl t† ˆ iˆ1 where the Zi,i = 1, . . .,m are de®ned in (18). Since the local performance output is parameterized in terms of pp and the membership functions are parameterized in terms of pm, the PDC inherits the same parameterization of the TS representation in terms of both pm and pp. 5.2.2.1. Optimal controllers. The state-feedback controller Ki for the i-th rule, is chosen to solve the following optimization problem o n min Ki stabilizing max kTei1 r k22 , kTei2 r k22 subject to kTeiF cmd 2 r k2 < ri , 21† where the nondimensional constant ri is a bound on the control e€ort, Tei1 r , Tei2 r and TeiF r are the transfer functions from r to ei1 , ei2 and eiFcmd , respectively, and cmd kTk2 indicates the standard H2-norm of the transfer function T de®ned by s   1 1  T jo†T jo† do : kTk2 : ˆ trace 2p ÿ1 The optimization problem (21) is a particular case of the generalized H2 (GH2) control problem [11]. The idea behind the choice of this class of controllers is given as follows. Although the overall performance is to be optimized for inputs of a ®xed form (the bumps in Eq. (4)), we want to have a suspension system that, when the inputs are not exactly as in Eq. (4), it has some guaranteed desirable property. The class of controllers proposed, in addition to achieve good bump re- 908 F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 sponses, has the following property. If the suspension travel j z1 j Rc1 (c1 de®ned in Eq. (12)), the active suspension system minimizes max t j x b t† j krk over all perturbations r of bounded energy krk, subject to max j Fcmd t† j Rkrkr1 13,000 N: t This roughly means that, if the perturbations are such that the suspension travel is `small', we minimize the worst-case car body acceleration peak for inputs with bounded energy. For details on the GH2 methodology see [11±13]. Stochastic interpretations of the solution to Eq. (21) can be found, for example, in [14]. Controllers solving Eq. (21) can be computed eciently by reducing the optimization problem to a linear matrix inequality (LMI) problem (see [11] and Appendix A). 5.2.3. GA-based parameter selection As stated before, the PDC is parameterized by pm and pp. Based on our knowledge of the control system, it is not dicult to select parameter vectors pm and pp that achieve a fairly good performance (our previous experience indicates this). However, to achieve an even better performance this intuitive approach is not good enough. In this section, we propose to use a GA [15] for selecting the ®nal pm and pp. The GA used for choosing pm and pp includes the standard processes for population initialization, reproduction, crossover, and mutation [15], and maximizes a ®tness function that captures the performance objectives. This ®tness function is given by J p m , pp † ˆ 1 , J1 ‡ J2 ‡ J3 22† with J1 ˆ 8 >  5† < 1500X J2 ˆ 8 > <0 b >  5† †4 : 1500 3X b  5† < if X b 1 3  5† r 1 if X b 3  11† R 3 if X b 2 >  11† r 3  11† ÿ 1† if X : 100 2 X b 3 b 2 23† 24† 909 F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 Fig. 5. Fuzzy membership functions for the controllers C2fs, C3fs and C4fs (abscissas [m]). J3 ˆ  106 0 if z1 ÿ Z1 < 10 ÿ4 z1 if z1 ÿ Z1 > 10 ÿ4 z1 , 25†  11† † is the ratio between the closed and open loop maximum peak of  5† X where X b b the body acceleration xÈb response to 5 cm bumps (11 cm bumps), and Z1 is the maximum peak of the suspension travel z1 for the closed loop response to 11 cm bumps. The rationale for selecting this ®tness function is as follows. The passenger comfort costs J1 and J2 penalize body accelerations for 5 cm bumps and for 11 cm Table 1 Controller C2fs Ð parameters pm (in cm) de®ning the membership functions a1 b1 c1 d1 ± ± 4.4539 7.5609 a2 b2 c2 d2 4.4539 7.5609 ± ± 910 F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 Table 2 Controller C2fs Ð parameters pp de®ning the performance outputs l1 l2 1 0 r1 r2 34.7462 37.3661 Fig. 6. Time traces for C2fs and 11 cm bump Ð solid line, closed loop; dashed line, open loop. 911 F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 Table 3 Controller C3fs Ð parameters pm (in cm) de®ning the membership functions a1 b1 c1 d1 ± ± 0.3983 2.3165 a2 b2 c2 d2 0.3983 2.3165 5.3997 6.8183 a3 b3 c3 d3 5.3997 6.8183 ± ± Table 4 Controller C3fs Ð parameters pp de®ning the performance outputs l1 l2 l3 1 0.9236 0 r1 r2 r3 39.5487 95.7383 28.2616 Table 5 Controller C4fs Ð parameters pm (in cm) de®ning the membership functions a1 b1 c1 d1 ± ± 1.4864 3.0646 a2 b2 c2 d2 1.4864 3.0646 5.0585 6.5626 a3 b3 c3 d3 5.0585 6.5626 7.3383 7.6669 a4 b4 c4 d4 7.3383 7.6669 ± ± 912 F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 Table 6 Controller C4fs Ð parameters pp de®ning the performance outputs l1 l2 1 0.4108 l3 0.9264 l4 0 r1 r2 r3 r4 43.5843 43.5843 61.5425 40.6836 Fig. 7. Time traces for C2fs and 5 cm bump Ð solid line, closed loop; dashed line, open loop. F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 913 bumps, respectively. The suspension lifetime cost J3 is used to discard controllers that hit the suspension travel limits with an 11 cm bump. The controller evaluation is performed via time-domain simulations. These simulations include the rattle-space saturation in Eq. (2), the linearizing controller in Eq. (5), and the actuator nonlinear dynamics in Eq. (3). 6. Results Using the proposed design methodology, we obtained three controllers: C2fs, C3fs and C4fs, which make use of 2, 3 and 4 fuzzy sets, respectively. The parameters de®ning these controllers are given in Tables 1±6, and the corresponding membership functions are plotted in Fig. 5. Fig. 8. Time traces for C3fs and 11 cm bump Ð solid line, closed loop; dashed line, open loop. 914 F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 Fig. 9. Time traces for C3fs and 5 cm bump Ð solid line, closed loop; dashed line, open loop. Figs. 6±11 show time traces of three representative variables for controller evaluation: (a) body acceleration xÈb, (b) suspension travel z1, and (c) actuator force F. All variables are given for 5 cm and 11 cm bumps. We observe that there is a substantial reduction in the acceleration peaks for 5 cm bumps, with respect to the open-loop response. At the same time, none of our designs hits the rattlespace limits with 11 cm bumps. The performance is measured using the following indices: a-: the maximum bump amplitude for which the suspension travel does not hit the rattle-space limits, and  0, 5† : the maximum ratio between closed/open-loop body acceleration peaks, X b for bumps with amplitude a R 5 cm. F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 915 Fig. 10. Time traces for C4fs and 11 cm bump Ð solid line, closed loop; dashed line, open loop. We perform time simulations, for a ®ne grid of bump amplitudes, to determine a 0, 5† : In Fig. 12 we plot acceleration ampli®cation as a function of the bump and X b amplitude a. From these plots it is evident the advantage of the fuzzy controller over linear controllers. Clearly, linear controllers cannot achieve variable performance depending on the disturbance amplitude.  0, 5† were 0.426, The resulting values of the acceleration ampli®cation X b 0.520 and 0.397 for controllers C2fs, C3fs and C4fs, respectively. The value of awas increased from 9.3 cm (for open-loop response) to 11 cm for all three controllers. As a result of the previous analysis performed on the three controllers C2fs, C3fs and C4fs, it is concluded that the controllers C2fs and C4fs, performed 916 F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 Fig. 11. Time traces for C4fs and 5 cm bump Ð solid line, closed loop; dashed line, open loop. approximately the same, and better than controller C3fs.3 Then, a good candidate for implementation is C2fs, since it has the least associated complexity. The achieved performance for this controller can be summarized by: With respect to open-loop, the passenger comfort was improved by 58% for small/ medium road disturbances, while the maximum bump amplitude for which the suspension limits are not hit was increased by 18%. 3 As the GA does not have guarantees to ®nd the global maximum of the ®tness function, it can not be claimed that the resulting controllers corresponding to a larger number of fuzzy sets will perform better. Then, performance has to be checked for all candidate controllers after running the GA optimization to select the best one. F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 917 Fig. 12. Evaluation criteria Ð Acceleration ampli®cation vs bump amplitude. Solid line: C2fs; dashed line: C3fs; dotted line: C4fs. 7. Conclusions An approach is presented for active suspension design which uses both inner and outer control loops. The inner loop controls the nonlinear hydraulic actuator in order to track the desired actuation force. The outer loop implements a fuzzy logic controller with parameters computed by GA-based optimization. The major contribution is a detailed design methodology to solve this particular class of problems. The approach combines di€erent control design techniques to achieve performance dependent on the perturbation size. The method consists in producing a class of controllers parameterized in terms of a small number of 918 F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 parameters, which are selected with the aid of a GA algorithm to obtain desired performance levels. Incorporation of other design objectives can be done in a straightforward manner. The methodology proved e€ective when applied to a quarter-car model of a suspension system. Further work is needed to address the robustness of the resulting controllers, and to extend the methodology to full-car models. Acknowledgements This work was supported in part by the National Science Foundation under Young Investigator Award ECS-9358288, and in part by Boeing Corporation, through the Control Systems Laboratory at Purdue University under the direction of Prof. Rotea. The authors would like to thank the reviewers for helpful suggestions. Appendix A. Solution to the optimization problem (21) Let the linear system z_l ˆ Az ‡ B1 r ‡ B2 Fcmd ei1 ˆ C1i z ‡ D1i Fcmd ei2 ˆ C2i z ‡ D2i Fcmd eiFcmd ˆ DFcmd i Fcmd , 26† be valid for the state-space region Xi , i ˆ 1, . . . , m; and consider controllers of the form Fcmdi ˆ Ki z: 27† Then, the closed-loop becomes z_l ˆ A ‡ B2 Ki †z ‡ B1 r ei1 ˆ C1i ‡ D1i Ki †zl ei2 ˆ C2i ‡ D2i Ki †zl eiu ˆ DFcmd i Ki zl : 28† F.J. D'Amato, D.E. Viassolo / Mechatronics 10 (2000) 897±920 919 The objective is to ®nd a stabilizing Ki that solves min Ki subject to d kTei1 r k22 < d 29† kTei2 r k22 < d kTeiF cmd 2 r k2 < ri , where pi > 0 is a given constant. (This problem is equivalent to the problem in Eq. (21).) Using Eq. (28) and results in [16], the problem in (29) can be written as min Q, Wi d subject to AQ ‡ QAT ‡ B2 Wi ‡ W Ti B T2 ‡ B1 B T1 < 0  dI C1i Q ‡ D1i Wi †T C1i Q ‡ D1i Wi Q  >0  dI C2i Q ‡ D2i Wi †T C2i Q ‡ D2i Wi Q  >0  ri I DFcmd i Wi †T DFcmd i Wi Q  > 0, 30† where Wi : ˆ Ki Q: This optimization problem is a linear matrix inequality (LMI) convex problem, which can be solved eciently [17,18]. References [1] Lin J-S, Kanellakopoulos I. Nonlinear design of active suspensions. IEEE Control System Magazine 1997;17:45±59. [2] Alleyne A, Hedrick JK. Nonlinear adaptive control of active suspensions. IEEE Trans Control Syst Technology 1995;3:94±101. [3] Alleyne A, Neuhaus PD, Hedrick JK. 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