Robust linear control of an active suspension on a quarter car test-rig

ARTICLE IN PRESS Control Engineering Practice 13 (25) 577 586 Robust linear control of an active suspension on a quarter car test-rig Christophe Lauwerys*, Jan Swevers, Paul Sas Mechanical Engineering, Division PMA, Katholieke Universiteit Leuven, Celestijnenlaan 3B, B-31 Heverlee, Belgium Received 16 September 23; accepted 26 April 24 Available online 2 June 24 Abstract This paper presents the design of a robust linear controller for an active suspension mounted in a quarter car test-rig. Unlike most active suspension control design methods, the presented approach does not require a (nonlinear) physical model of either the car or the shock-absorber, which are very time consuming to derive. The presented method is based on linear techniques well supported by CACSD-software tools, yielding a fast control design approach, applicable to almost any active suspension system. Linear black box models are identified using frequency domain identification techniques, while robust linear control design techniques (H N and m- synthesis) account for the model uncertainties introduced by the linear model approximation of the nonlinear dynamics. Although this linear approach introduces some conservatism, it is shown that the desired performance is achieved in simulation as well as on the experimental test-rig. r 24 Elsevier Ltd. All rights reserved. Keywords: Active vehicle suspension; Robust control; Structured singular value 1. Introduction Comfort and road-handling performance of a passenger car are mainly determined by the damping characteristic of the shock absorbers. Passive shock absorbers have a fixed damping characteristic determined by their design. Depending on the road excitation, however, it is desirable to adjust this characteristic to increase performance. Semi-active and active suspension systems offer the possibility to vary the damper characteristics along with the road profile e.g. by changing the restriction of one or two current controlled s. An active shock absorber has the additional advantages that negative damping can be provided and that a larger range of forces can be generated at low velocities, thereby potentially allowing an increase in system performance. Literature describes several linear and nonlinear techniques to control a car using an active suspension. Karnopp and Crosby (1974). Butsuen, (1989) and Esmailzadeh and Taghirad (1995) apply linear control *Corresponding author. Tel.: +32-163-228-32; fax: +32-163-229-87. E-mail address: christophe.lauwerys@mech.kuleuven.ac.be (C. Lauwerys). strategies based on linear physical car models consisting of lumped masses, linear springs and dampers, and an active shock absorber modelled as an ideal force source. Real car dynamics are much more complex and active shock absorbers are not ideal force sources but have a complex nonlinear dynamic behaviour. As a result of these unrealistic assumptions, these linear control approaches are not appropriate for practical applications. Nonlinear control strategies such as linear parameter varying gain scheduling (Fialho & Balas, 22), (Sz!aszi, 22) and backstepping (Lin & Kannellakopoulos, 1997), have been applied to active suspension systems and validated by means of simulations only. These controllers are based on a linear or nonlinear, physical car model in combination with a nonlinear physical damper model. These models have a large number of parameters. The experimental identification of these model parameters is a complex (non-convex optimization) problem. In addition, design and tuning of the above-mentioned nonlinear controllers are not straightforward. Basically, the use of nonlinear models and controllers leads to very time-consuming designs, since no standard techniques or software tools are available. Finally, the implementation of these controllers is too complex for practical use in a passenger car control system. 967-661/$ - see front matter r 24 Elsevier Ltd. All rights reserved. doi:1.116/j.conengprac.24.4.18

578 ARTICLE IN PRESS C. Lauwerys et al. / Control Engineering Practice 13 (25) 577 586 This paper discusses the application, implementation and validation of an integrated experimental linear robust control design approach on a quarter car test-rig equipped with an active suspension system. This approach is guided by practical considerations with respect to controller complexity and ease of design and tuning. It is based on linear experimental identification and robust linear control design techniques, supported by CACSD-tools, available e.g. in Matlab. The system is excited and a linear model is identified based on the measured frequency response. The modelling error caused by noise, nonlinearities and neglected dynamics are used to estimate the model uncertainty. A robust linear controller is designed using m-synthesis. All steps in these identification and control design procedures are time-efficient compared to the corresponding nonlinear techniques. However, since the conservativeness of the resulting controller depends on the linearity of the system, it was not certain in advance whether this approach could be successful on the considered system which contains elements, e.g. the active shock absorber, that are highly nonlinear. The experimental results discussed in this paper show, however, that the dynamic behaviour of the considered quarter car test-rig is indeed sufficiently linear to achieve the required performance using a linear controller. This aspect is validated by comparing the measured and predicted closed-loop performance. Elements of robust control design used in this paper are briefly reviewed in Section 2. Section 3 gives the description of the quarter car test set-up and the active suspension. Section 4 discusses the experimental identification of the system and estimation of the model uncertainty. The design and experimental validation of the robust controller are discussed in Sections 5 and 6, respectively. Final conclusions are given in Section 7. (or perturbed) plants G p defined by an uncertainty model. Define RP 3 jjpjj N p1 8G p AP: ð2þ To check robust performance using the structured singular value m; the model uncertainty has to be modelled as norm-bounded perturbations D i on the nominal system. The perturbations are collected in a block-diagonal matrix D U ¼ diagðd 1 ; y; D n Þ such that the closed-loop system can be represented as in Fig. 1 and the robust performance condition reduces to RP 3 m DU ðf L ðp; KÞÞp1; where F L ðp; KÞ is the lower linear fractional transformation of the system and the controller, defined as F L ðp; KÞ ¼P 11 þ P 12 KðI P 22 KÞ 1 P 21 : 2.2. Representing uncertainty The quarter car test-rig with the active shock absorber is a complex nonlinear system. For control design, however, it is desirable to have a simple (low-order), linear model describing the most important dynamics of the system. Model accuracy and simplicity are conflicting criteria in the identification procedure. Linear approximation of the plant introduces model uncertainty, which is described here as multiplicative uncertainty W T ; which is calculated as the frequency domain representation of the worst-case relative difference between the model G and elements G p of P: W T ðoþ ¼ max G p ðoþap G p ðoþ GðoÞ GðoÞ : These elements G p can be obtained as frequency response functions (FRFs) measured on the system. The set P is then described as ð3þ ð4þ 2. Robust linear control 2.1. Robust performance P : G p ¼ Gð1 þ W T DÞ ð5þ The controller requirements that the closed-loop system is stable and satisfies some performance criteria for the nominal plant G are called nominal stability (NS) and nominal performance (NP), respectively (Skokestad & Postlethwaite, 1986). In H N control, the performance criteria are defined using the H-infinity norm (Skokestad & Postlethwaite, 1986). Define NP 3 jjpjj N p1; ð1þ where P is obtained by augmenting the nominal plant G with weighing functions that describe the required performance. The term robust performance (RP) indicates that these requirements are satisfied for a set P of possible u w u U P K Fig. 1. General control configuration for a model including uncertainty. v z v

ARTICLE IN PRESS C. Lauwerys et al. / Control Engineering Practice 13 (25) 577 586 579 W T G p + rod cylinder + G Fv piston Fv Fig. 2. Plant with multiplicative uncertainty. and represented as shown in Fig. 2. The estimation of G and W T for the considered system are discussed in Section 4. 3. Description of the quarter car test set-up The quarter car test-rig consists of an active shock absorber, described in Section 3.1, and the quarter car structure, described in Section 3.2. 3.1. Description of the active shock absorber The active shock absorber hardware is designed such that its open-loop (uncontrolled) dynamic characteristic is comparable to that of a passive shock absorber tuned for the same type of car. The performance is then further improved by means of a controller that regulates the damping characteristic of the active shock absorber according to the road. A classic passive shock absorber consists of a cylinder filled with oil and a rod connected to a piston, which contains a calibrated restriction called the piston (see Fig. 3 left). The change in volume caused by the rod moving in or out of the cylinder is compensated by oil flowing in or out of the accumulator (accu) through the base. The pressure drop over both the base and the piston results in a damping force acting on the piston. The active shock absorber hardware (see Fig. 3 right) corresponds to that of a passive shock absorber in which the piston and base are each replaced by a check and a current-controlled CVSA 1. The current to this is limited between i ¼ :3A and i þ ¼ 1:6A; which corresponds to the least and most restrictive positions of the (i.e. open and closed), respectively. When the rod moves up (positive velocity), the piston check- closes and oil flows through the piston CVSA. Because the volume of the rod inside the cylinder reduces, oil is forced from the accumulator into the cylinder through the base check-, rendering the restriction of the base CVSA unimportant. For positive velocities, the damping force is thus mainly controlled by the current to the piston CVSA i p (see Fig. 4). 1 CVSA: continually variable semi-active. accu Damping force (N) 6 4 2-2 -4-6 piston base accu i b =i +,i p =i -,i v =+1 i b =i -,i p =i +,i v =-1 piston CVSA base CVSA piston check base check Fig. 3. Passive (left) and active (right) shock absorber schemes. i b =i -,i p =i -,i v = o pump -1 -.8 -.6 -.4 -.2.2.4.6.8 1 Rattle velocity (m/s) Fig. 4. Feasible working range of the active shock absorber. By varying only one control current at the time, the entire working range is covered. The velocity is positive when the rod is moving out of the cylinder, the force is positive when the damper pushes the rod out of the cylinder. When the rod moves down (negative velocity), the piston check- opens, rendering the restriction of the piston CVSA unimportant. Because the volume of the rod inside the cylinder increases, the base check closes and oil flows from the cylinder into the accumulator through the base CVSA. For negative velocities, the damping force is thus mainly controlled by the current to the base CVSA i b (see Fig. 4). The active performance of the system is realized by adding to the shock-absorber a hydraulic pump generating constant flow. Fig. 4 shows the simulated feasible working range of the active shock absorber. The damping force is plotted as a function of the rattle velocity 2 for several combinations of currents to the base and piston, 2 Rattle velocity: relative velocity of the rod with respect to the cylinder; positive values corresponding to the rod moving out of the cylinder.

58 ARTICLE IN PRESS C. Lauwerys et al. / Control Engineering Practice 13 (25) 577 586 respectively, i b and i p : The area of feasible forces is divided into two domains by the force velocity curve obtained when i b ¼ i p ¼ i : The upper half of the working range is obtained by increasing the current to the base from i to i þ ; while keeping the current to the piston at i : The lower half of the working range is obtained by increasing the current to the piston from i to i þ while keeping the current to the base at i : The fact that the entire envelope is covered by varying only one current signal at the time is exploited by introducing a new, artificial signal i v ; a virtual current, which varies from 1 to +1 and corresponds to the following specific combination of base and piston currents i b ¼ i i v ði þ i Þ; i v o; i ; i v > ; i p ¼ i ; i v o; i þ i v ði þ i Þ; i v > : spring tire stiffness body i v wheel rod road bushing active damper a w Fig. 6. Schematic representation of the system. x a a b This single virtual current replaces the two original currents to the base and the piston as the system input signal in the control design. This substitution reduces the controller from a multiple-output to a single-output system which is clearly more straightforward to design. x a i v G 11 G 21 G 12 G 22 a b a w K 11 3.2. Description of the quarter car structure The quarter car structure (see Figs. 5 and 6) is a testrig built to simulate the vertical dynamics of a single corner of a car. It consists of a wheel, a sliding cantilever representing one quarter of a car body mass, and in between these two, a spring and an active damper with bushing. The wheel is excited at the bottom by a hydraulic actuator thereby simulating a vertical road K 12 Fig. 7. Block diagram of the system. displacement. A guideway imposes a vertical motion on the beam and the wheel. Fig. 6 shows a schematic representation of this system. From a control design point of view, this is a multi-input multi-output (MIMO) system. The acceleration of the body and the wheel a b and a w are the outputs of the system available to the controller. The virtual current i v to the active shock absorber is the input of the system calculated by the controller. The imposed road displacement x a acts on the system as an external disturbance. Fig. 7 represents these input output relations in a schematic way. The system dynamics are represented by the 2 2 transfer matrix G 22 ; the controller by the 1 2 transfer matrix K 12 : 4. System identification Fig. 5. Picture of the quarter car test-rig equipped with the active suspension. The system dynamics and uncertainties are identified using a frequency domain approach, consisting of four steps: (1) design of appropriate excitation signals, (2)

ARTICLE IN PRESS C. Lauwerys et al. / Control Engineering Practice 13 (25) 577 586 581 estimation of the FRFs of the system, (3) identification of a parametric nominal model for these FRFs, and (4) estimation of the model uncertainty. 4.1. Experiment design In order to identify the dynamics of this system, excitation signals have to be designed for the inputs i v and x a : A common rule in identification is to use excitation signals that correspond to the realistic excitation of the system, such that the identified linear model is a good approximation of the system for that type of excitation (Schoukens & Pintelon, 21). The control signal i v to the active shock absorber is excited with Gaussian band-limited white noise. The bandwidth is set to 5 Hz; which is sufficiently higher then the desired closed-loop bandwidth. The amplitude level is chosen such that the 2s-bound of the excitation signal corresponds to the saturation levels 71 to get a good signal-to-noise ratio (Bendat & Piersol, 2). The road displacement signal x a generated by the hydraulic shaker is excited with integrated Gaussian white-noise, which spectrum corresponds to that of a stochastic road. The measured outputs are the acceleration of the body a b and wheel a w : 4.2. FRF matrix estimation The excitation signals are applied simultaneously to the system for 3 s sampled at 1 khz: The first resonance of the system is the well-damped body mode and is expected at 1:5 Hz: The lowest frequency of interest is chosen approximately a factor five smaller, leading to a frequency resolution of :24 Hz: The measurement data is divided into 145 blocks of 496 samples, with 5% overlap. An H 1 -estimator (Bendat & Piersol, 2) is used to estimate the MIMO FRF matrix (see Fig. 8, dotted lines). 4.3. Identification of parametric transfer function (TF) matrix G A parametric TF matrix is fitted on the measured MIMO FRF matrix, using the nonlinear least squares frequency domain identification method (Schoukens & Pintelon, 21). The aim is to fit a simple yet accurate linear MIMO model on the measured MIMO FRF matrix. The convergence of this optimization is improved if a good estimate of the number of model parameters (number of system poles and TF zeros) is available, and if a priori system knowledge based on physical insight can be included. 1 5-5 1-1 1 1 1 1 2 4 2-2 -4 1-1 1 1 1 1 2 2 1-1 -2 1-1 1 1 1 1 2 2 1-1 -2 1-1 1 1 1 1 2 1 5-5 1-1 1 1 1 1 2 5-5 -1 1-1 1 1 1 1 2 2 2 1 1-1 -1-2 -2 1-1 1 1 1 1 2 1-1 1 1 1 1 2 Fig. 8. FRFs of G measured (dotted) and calculated from the reduced order MIMO model (solid): from road displacement x a (left) and virtual current i v (right) to body (top) and wheel (bottom) acceleration a b and a w :

582 ARTICLE IN PRESS C. Lauwerys et al. / Control Engineering Practice 13 (25) 577 586 This information is obtained by representing the system as a lumped parameter model consisting of three masses connected by springs and dampers, as shown in Fig. 6. The dynamics of the active shock absorber are for this purpose neglected by assuming that the shock absorber force is proportional to the control current. These assumptions suggests that a linear model of order six exists which describes the three system modes: the body mode, the wheelhop mode and a bushing mode. Analysis of this lumped parameter model also shows that a double differentiator (double zero at Hz) is included in all four transfer functions. The nonlinear least-squares identification of a single MIMO model on the measured MIMO FRF matrix did, however, not converge to a useful solution. The bad quality of the initial estimate of this complex nonconvex optimization problem yielded convergence to a local minimum instead of the global one. Therefore, an alternative approach was followed. A single-input single-output (SISO) TF model of sixth order was fitted on each FRF separately. The three complex pole pairs, representing the three system modes, should, in theory, coincide for all four SISO models. However, due to noise and nonlinearities the poles of the individual models differ slightly. Some kind of model reduction has to be applied while combining these models to avoid illconditioned optimization in the control design and to reduce the complexity of the controller. In order to accomplish this model reduction, the poles of the four SISO TFs are changed to the average values of the poles of the identified SISO models. The zeros are re-identified in the frequency domain in order to fit these new common-pole models on the measured FRFs. This reidentification is a linear least squares problem, which encounters no convergence problems. The obtained reduced order MIMO model is only slightly less accurate (in a least squares sense) than the MIMO model obtained by combining the four SISO model without reducing the number of common poles. The system also contains a delay of 6 ms; originating from the hydraulic tubing and the electrodynamic s. It was estimated based on the linear-phase lag present in all FRFs. A second-order Pade approximation is used to convert this delay to a pair of complex poles and right-half-plane zeros. These poles and zeros were added to the reduced-order model, yielding the parametric nominal model G (see Fig. 7). 4.4. Estimation of linear multiplicative uncertainty models The nominal model G is a linear approximation of the quarter car test-rig dynamics. Model uncertainties are caused by sensor noise, nonlinearities and unmodelled high-frequency dynamics. The multiplicative uncertainties W ab T and Waw T (see Eqs. (4) and (5) in Section 2.2) are 5 4 3 2 1-1 -2 1-1 1 1 1 1 2 5 4 3 2 1-1 -2 1-1 1 1 1 1 2 Fig. 9. FRFs of the estimated multiplicative uncertainty (dotted) and the fitted linear models (solid). estimated as the maximum of the relative error between the measured FRFS and the identified nominal model (see Fig. 9). A twelth-order model is fitted on the multiplicative uncertainty for the body acceleration output (see Fig. 9 top). The uncertainty is well below 1% ( db) between :5 and 1 Hz: The lightly damped zero at 12 Hz in the model G causes the relative uncertainty to peak at this frequency. A fourth-order model is fitted on the multiplicative uncertainty for the wheel acceleration output (see Fig. 9 bottom). The uncertainty is below 1% ( db) between 1 and 15 Hz: The model orders where chosen such that the obtained models fitted the data sufficiently accurately in the frequency range of interest. 5. Control design The aim is to design a linear controller that attenuates the body acceleration in the frequency region around the

ARTICLE IN PRESS C. Lauwerys et al. / Control Engineering Practice 13 (25) 577 586 583 body mode ð1:5 HzÞ; without amplifying the body acceleration or tire force variations in other frequency regions. These specifications correspond to improving both the comfort and handling characteristics. Robust performance is accounted for by taking into account estimated model uncertainties. A standard passive shock absorber tuned for this suspension system is used as a reference (see Fig. 1). 5.1. Design of weighting functions The generalized plant P used in classic robust control design (see Eq. (3) and Fig. 1) consists of two types of inputs and outputs: exogenous inputs and outputs, and controller inputs and outputs. The frequency content of the exogenous inputs signals and the desired frequency content of the exogenous output signals is expressed by frequency domain weighting functions, such that the 3 25 2 15 1 5-5 -1 8 78 1 1 1 desired performance can be expressed as a bound on the H-infinity norm of the augmented plant. This section explains how this augmented plant is obtained for the quarter car system with the active suspension (see Fig. 11). The system has one exogenous input, the simulated road displacement x a ; which acts as an unmeasured disturbance. The frequency content of a typical stochastic road is integrated white noise, which is modelled by the weighing function W r : It has a lag-compensator characteristic with a 2 db=decade slope in the frequency region of interest, and a zero slope at low and high frequencies to satisfy the necessary conditions for a sound state-space H-infinity control design problem formulation (Skokestad & Postlethwaite, 1986). A first exogenous output signal z u is defined as the weighted control input i v ; using a weighting function W u : W u is set equal to one, expressing that the two-norm of the control input should be smaller than one for all frequencies to avoid excessive saturation of the control current i v : A second exogenous output signal z P is defined as the weighted body acceleration signal a b ; using a weighting function W P : W P is a band-pass filter with cut-off frequencies around the body mode resonance, such that the amplitude of the system from w r to z P is larger than one in the frequency region of this resonance. This expresses that the controller should attenuate the body acceleration at the body mode resonance. The exact locations of the cut-off frequencies of this band-pass filter weight were the tuning parameters of this control design. The desired width of the frequency region of attenuation was increased until the control design problem became infeasible. The system is also augmented with the multiplicative output uncertainty models W ab T and Waw T : This creates the uncertainty inputs u ab D u aw D : and uaw D and outputs vab and D 76 74 W u z u P1 72 7 68 66 64 w r W r x a i v G 11 G 21 G 12 G 22 a b a w W P ab W T aw W T z P v ab v aw P2 U1 U2 u ab u aw 62 6 1 1 1 Fig. 1. FRFs from road to body acceleration (top) and tire force (bottom) for the uncontrolled active suspension (solid) and the tuned passive suspension (dashdot). K 11 K 12 Fig. 11. Generalized plant for the quarter car system with active suspension.

584 ARTICLE IN PRESS C. Lauwerys et al. / Control Engineering Practice 13 (25) 577 586 The generalized plant P is obtained by absorbing the weights W r ; W u ; W P ; W ab T and Waw T in the model G yielding the block diagram representation of Fig. 12. 5.2. H N design and m-synthesis A controller K achieves robust performance if it stabilizes the closed-loop system shown in Fig. 12. The H-infinity control design framework is conservative in that it cannot take into account the structure of the D- blocks. Using this framework it was impossible to find a stabilizing controller. The m-synthesis control design framework is capable of taking into account the structure of the D-blocks since it is based on the structured singular value of the closedloop system (see Eq. (3)). The controller synthesis is based on the DK-iteration scheme (Skokestad & Postlethwaite, 1986) in which alternatingly D-scaling functions are fit to approximate the structured singular value (D-step) and an H-infinity controller is designed (K-step). Fig. 13 shows the synthesized controller obtained with m-synthesis. The inputs of the controller are the measured body and wheel acceleration a b and a w ; the output is the virtual current to the active shock absorber i v : The order of the controller is 6, which is the order of the augmented plant plus the order of the D-scalings. 5.3. Controller order reduction P1 P1 The high order of the controller has a number of disadvantages briefly mentioned below: u1 * Numerically ill-conditioned state space matrices. * Real time processor overload. * Excessive memory usage. u2 u,w P 11 P 12 v,z u P 21 P 22 v Therefore, the order of the controller is reduced by an optimal Hankel approximation (Skokestad & Postlethwaite, 1986). Fig. 14 shows the Hankel singular value spectrum of the controller and indicates the selected order: 14. Comparison of the FRFs of the reduced order and the original controller in Fig. 13 shows that there are only small deviations at higher frequencies above 2 Hz such that deterioration of the closed-loop performance is not expected. K 11 K 12 Fig. 12. Generalized plant with absorbed weights. 6. Controller validation The performance of the controller is validated by comparing the open loop with the closed-loop disturbance TFs from the road to the body acceleration. This validation is performed both in simulation based -1-2 -3-2 -4 1 1 1 1 2-6 1 1 1 1 2 5-5 1 1 1 1 2 2 15 1 5 1 1 1 1 2 Fig. 13. Controller obtained with m-synthesis (solid) and reduced-order controller (dotted). The controller inputs are the body acceleration a b (left) and the wheel acceleration a w (right). The controller output is the virtual current i v :

ARTICLE IN PRESS C. Lauwerys et al. / Control Engineering Practice 13 (25) 577 586 585 26 24 22 2 18 16 14 12 1 Fig. 14. Hankel singular value spectrum of the controller. on the identified linear models, and experimentally on the quarter car test-rig. 6.1. Simulated closed-loop performance The open- and closed-loop disturbance TFs from the road to the body acceleration are calculated analytically based on the identified linear models. Fig. 15 shows that the controller attenuates the body acceleration around the body mode, from 1 up to 5 Hz; by 6 db or 5% of its uncontrolled value. At higher frequencies (above 2 Hz) there is only a slight amplification w.r.t. the uncontrolled disturbance TF. 6.2. Experimental closed-loop performance The hydraulic shaker that imposes a desired road profile to the quarter car is excited with a typical stochastic road displacement profile (integrated white noise) while the body and wheel acceleration are measured. Three suspensions were compared: a standard passive shock absorber tuned for this set-up, the active shock absorber used in open loop (i.e. with no feedback control), and the active shock absorber with the m-synthesis controller. The performance with respect to comfort is compared by considering the disturbance transfer functions from the road to the body acceleration (see Fig. 16). Comparing performance for the uncontrolled and controlled active shock absorber shows that the controller attenuates the body mode resonance with 6 db as was expected from the simulation results. An additional resonance occurs at 2 Hz but its amplitude remains below that of the other resonances. Comparing the tuned passive shock absorber with the controlled active shock absorber shows that the body acceleration is attenuated with 6 db around the body mode resonance, while it remains attenuated up to 6 Hz: 8 1 1 1 Fig. 15. Comparison of the simulated disturbance FRFs from the road to the body acceleration for the uncontrolled (solid) and controlled (dotted) active suspension. 3 25 2 15 1 5 1 1 1 Fig. 16. Comparison of the experimental disturbance FRFs from the road to the body acceleration for the uncontrolled (solid) and controlled (dotted) active suspension, and for the tuned passive suspension (dashdot). The performance with respect to road handling was not taken into account explicitly in the control design. However, it is validated experimentally by comparing the peak value of the disturbance TFs from the road displacement to the tire force (see Fig. 17). Comparing the open-loop with the closed-loop response of the active shock absorber shows that the controller attenuates the tire force around the body mode resonance, while it amplifies the tire force around the wheelhop mode ð12 HzÞ: However, the peak value of the tire force for the controlled active suspension remains below the peak value for the tuned passive suspension. Therefore, the performance with respect to road handling is still

586 ARTICLE IN PRESS C. Lauwerys et al. / Control Engineering Practice 13 (25) 577 586 8 78 76 74 72 7 68 66 64 62 6 1 1 1 Fig. 17. Comparison of the experimental disturbance FRFs from the road to the tire force for the uncontrolled (solid) and controlled (dotted) active suspension, and for the tuned passive suspension. appropriately designed experiments. m-synthesis based on the DK-iteration scheme was used to design the controller, taking into account the estimated model uncertainties and constraints on the input current to the s. The inherent high order of the controller was substantially reduced by using an optimal Hankel approximation without deterioration of the closed-loop performance. The nominal model and the uncertainty models are identified sufficiently accurately, since the simulated results correspond closely to the experimental results. This indicates that this approach could be used successfully on full vehicles with similar active suspensions as well. Another advantage of this integrated control design approach is that it is fast, because it relies on linear black-box techniques well supported by CACSD-software tools, e.g. Matlab. This avoids the time consuming task of deriving, physical, often nonlinear, models of either the car or the shock-absorber. better for the controlled active suspension compared to the tuned passive suspension. The order of the resulting controller is still rather high (14th order), which is a consequence of the used robust control design technique that naturally integrates the desired performance specifications and model uncertainties expressed in the frequency domain. This performance and robustness can be approximated using a classical controller with a limited number of parameters. However, while the presented robustly performant controller was obtained automatically using m-synthesis techniques, the tuning of classical controllers has to be performed manually and iteratively since robustness can only be verified afterwards. 7. Conclusions The application of linear robust control design techniques on a quarter car test-rig equipped with an active suspension resulted in a controller that is able to significantly increase comfort compared to the comfort obtained with a tuned passive shock-absorber: the attenuation of the body acceleration around the body mode frequency is increased with 5%. The obtained handling performance is comparable to that of the tuned passive shock absorber: there is no significant increase of peak values of the tire force. The presented approach includes the experimental frequency domain identification of linear models of the quarter car system, including the active shock absorber, and the estimation of the model uncertainties introduced by this linearization, based on data obtained from Acknowledgements The financial support by the Flemish Institute for Support of Scientific and Technological Research in Industry (IWT) is gratefully acknowledged. This research is sponsored by the Belgian programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister s Office, Science Policy Programming (IUAP). References Bendat, J. S., & Piersol, A. G. (2). Random data. New York: Wiley. Butsuen, T. (1989). The design of semi-active suspensions for automotive vehicles. Ph.D. thesis, Massachusetts Institute of Technology. Esmailzadeh, E., & Taghirad, H. (1995). State-feedback control for passenger ride dynamics. Transactions of the Canadian Society for Mechanical Engineering, 19(4), 495 58. Fialho, I., & Balas, G. J. (22). Road adaptive active suspension design using linear parameter varying gain-scheduling. IEEE Transactions on Control Systems Technology, 1(1), 43 54. Karnopp, D. C., & Crosby, M. J. (1974). Vibration control using semiactive force generators. Journal of Engineering for Industry, 96, 619 626. Lin, J.-S., & Kannellakopoulos, I. (1997). Road adaptive nonlinear design of active suspensions. Proceedings of the American Control Conference (pp. 714 718). Schoukens, J., & Pintelon, R. (21). System identification: A frequency domain approach. Piscataway, NJ: IEEE Press (8855-1331). Skokestad, S., & Postlethwaite, I. (1986). Multivariable feedback control analysis and design. New York: Wiley. Sz!aszi, I.P., & G!asp!ar, J.B. (22). Nonlinear active suspension modeling using linear parameter varying approach. Proceedings of the Tenth IEEE Mediterranean Conference on Control and Automation, MED22, Portugal THA (pp. 5 4).