Elsevier

Control Engineering Practice

Volume 16, Issue 12, December 2008, Pages 1519-1534
Control Engineering Practice

A new semi-active suspension control strategy through LPV technique

https://doi.org/10.1016/j.conengprac.2008.05.002Get rights and content

Abstract

This paper is concerned with the design and the analysis of a semi-active suspension controller. In the recent years different kinds of semi-active control strategies, like two-state Skyhook, LQ-clipped or model-predictive control, have already been developed in the literature. In this paper, a new semi-active suspension control strategy that a priori satisfies the principal limitations of a semi-active suspension actuator (dissipative constraint and force bounds) is introduced using the tools of the linear parameter varying (LPV) theory. This new approach exhibits some interesting advantages (implementation, performance flexibility, robustness, etc.) compared to already existing methods. Both industrial criterion based evaluation and simulations on a nonlinear quarter vehicle model are performed to show the efficiency of the method and to validate the theoretical approach.

Introduction

Global vehicle control is now an important issue in the future of vehicle control technology (see Shibahata, 2005). Many systems are involved in cars to guarantee both safety (e.g. ABS, ESP, etc.) (see, e.g., Corno, Savaresi, Tanelli, & Fabbri, 2008) and comfort (e.g. suspensions, seats, etc.). Concerning vehicle comfort and road holding in normal cruise situation (i.e. not in emergency case like over-steering or rolling situations), it is well admitted that controlled suspension systems can provide good passenger isolation from road unevenness while keeping admissible the road holding performances. Industrial and academic researches are very active in the automotive field; suspension design and control are important aspects for comfort and safety achievements (see Fischer & Isermann, 2003; Giorgeou, Verros, & Natsiavas, 2007; Hrovat, 1997; Ieluzzi, Turco, & Montiglio, 2005; Kawabe, Isobe, Watanabe, Hanba, & Miyasato, 1998).

In this area, an extensive literature and many models are available and widely used. In the last decade, many different active suspension system control approaches were developed. The Skyhook control (see Poussot-Vassal, Sename, Dugard, Ramirez-Mendoza, & Flores, 2006; Sammier, Sename, & Dugard, 2003), suits well to improve comfort but is limited to improve road holding. The linear quadratic control (see Hrovat, 1997) can provide both comfort and road holding improvements but requires the full state measurement or estimate. Linear time invariant (LTI) H control (see Rossi & Lucente, 2004; Sammier et al., 2003; Zin, Sename, & Dugard, 2005) can achieve better results improving both comfort and road holding ensuring a defined frequency behavior but is limited to fixed performances (due to fixed weights). Mixed LTI H/H2 approaches (see Abdellahi, Mehdi, & Saad, 2000; Gáspár, Szaszi, & Bokor, 1998; Lu, 2004; Lu & DePoyster, 2002; Poussot-Vassal et al., 2006; Takahashi, Camino, Zampieri, & Peres, 1998; Tuan, Apkarian, & Hosoe, 2001) that can improve H control reducing signals energy. Recently, the use of linear parameter varying (LPV) approaches (see Fialho & Balas, 2002; Gáspár, Szaszi, & Bokor, 2004; Poussot-Vassal, Drivet, Sename, Dugard, & Ramirez-Mendoza, 2007) leads to controllers that can either adapt the performances according to measured signals (road, deflection, etc.) or improve robustness, taking account of the nonlinearities (see Poussot-Vassal et al., 2006; Zin, 2005; Zin et al., 2006, Zin et al., 2008).

Most of these controllers are designed and validated assuming that the actuator of the suspension is fully active. Unfortunately, such active actuators are not yet used on a wide range of vehicles because of their inherent cost (e.g. energy, weight, volume, price, etc.) and low performance (e.g. time response); hence, in the industry, semi-active actuators (e.g. controlled dampers) are often preferred. Even if efforts on bounding the actuator force have been done using time domain (e.g. Chen & Guo, 2005) or frequency domain (e.g. Gáspár et al., 2004) constraint, the controller does not fulfill the dissipative constraint of the actuator. Hence, the performances obtained in simulation will be lost after implementation due to the controlled damper structural limitations. Usually, the method is to build an active controller and then, make it semi-active by saturating the forces required (known as the “clipped” approach), which does not ensure internal stability and performances any longer.

In the last years, different kinds of strategies have been developed to tackle such limitations, but semi-active suspension control remains an open research area. The two-state Skyhook control is an on/off strategy that switches between high and low damping coefficients in order to achieve body comfort specifications. Mixed Skyhook and ADD (see Savaresi, Silani, & Bittanti, 2004; Savaresi & Spelta, 2007) is also a comfort oriented control strategy involving switching strategy. Clipped approaches lead to unpredictable behaviors and reduce the achievable performances (see Canale, Milanese, & Novara, 2006; Giorgetti, Bemporad, Tseng, & Hrovat, 2006). In Giorgetti et al. (2006), authors compare different semi-active strategies based on optimal control and introduce a hybrid model-predictive optimal controller. The resulting control law is implemented via a hybrid controller that switches between a large number of controllers (function of the prediction horizon) and requires a full state measurement. In Canale et al. (2006), another model-predictive (MPC) semi-active suspension is proposed and results in good performances compared to the Skyhook and LQ-clipped approaches but it requires an on-line “fast” optimization procedure. As it involves optimal control, full state measurement and a good knowledge of the model parameters are necessary (see also Giua, Melas, Seatzu, & Usai, 2004).

The contribution of this paper is to propose new results to design a semi-active suspension controller. The methodology is developed for the first time through the LPV theory which allows to ensure internal stability and some performance criteria for the semi-active suspension. The main interest of such an approach is that it a priori allows to fulfill the dissipative actuator constraint and enable the designer to build a controller in the robust framework (H, H2, mixed, etc.) which is appreciated in the industry because of its flexibility (due to the weight functions) and its robustness to modeling uncertainties. As long as the new methodology does not involve any on-line optimization process and only requires a single sensor, it could be an interesting algorithm from the application's point of view. The proposed methodology is illustrated on a semi-active suspension equipped with a magneto-rheological (M-R) damper. The authors stress that the proposed design can be applied to any semi-active suspension actuators (not restricted to M-R dampers).

The paper is organized as follows. Section 2 introduces the nonlinear quarter car models used for synthesis, performance evaluation and simulation. In Section 3, the involved semi-active suspension actuator system (based on real experimental data) is described. In Section 4, a quarter vehicle frequency based industrial criterion is introduced in order to tune and evaluate the control strategy. Section 5 is devoted to recall the background and some facts on the LPV theory. In Section 6, the proposed semi-active LPV/H control design and its scheduling strategy are presented. In Section 7, both performance evaluation using the introduced criterion and time domain simulations are done on a nonlinear quarter vehicle model to show the efficiency of the proposed method. Conclusions on this new semi-active suspension approach and perspectives are discussed in Section 8.

Section snippets

Quarter car model

In the sequel, the involved passive reference and controlled nonlinear quarter vehicle models are described. Such a quarter car model is suitable when suspension control has to be synthesized. It makes possible to analyze the vertical behavior.

Semi-active suspension actuator

In the previous section, the control input u was introduced to control the quarter car model. Since here the focus is done on semi-active suspension control, in the sequel, emphasis is put on static performances and structural limitations for the considered semi-active actuator.

Performance objectives and evaluation

In this section both nonlinear frequency computation and performance evaluation are described. Controller tuning and closed-loop evaluation are done (in Section 6) according to this frequency based performance measure.

Background on LPV systems and gain-scheduled H polytopic controller

Before introducing the main result of this paper, which uses well known LPV and robust results, let us first recall some basic facts on LMI based H problem resolution for LPV systems.

Thanks to recent works on robust linear control (see Apkarian & Gahinet, 1995; Chilali & Gahinet, 1996; Chilali, Gahinet, & Apkarian, 1999; de Souza & Trofino, 2005; Gahinet, Apkarian, & Chilali, 1996; Scherer, 2004; Scherer, Gahinet, & Chilali, 1997) and references therein, LMI and optimization tools (see, e.g.

Main result: LPV based robust semi-active suspension control design

The controller synthesis is based on the model described in (1) where Fk(zdef)=k.zdef and Fc(z˙def)=c.z˙def are linear functions (c and k are given in Table 1). The control law, applied on model (2), is then given byu=-c(z˙def)+uHwhere c is the nominal linearized damping coefficient of the considered semi-active damper (for the M-R damper, c nominal is the one given to provide nominal performances, e.g. no current is applied to the system) and uH the additional force provided by the

Time and frequency results

In this section a validation of the proposed control approach is performed. First, performance evaluation, based on the previously introduced criterion, is done to show the control improvement. Then time domain simulations are performed to check that the semi-active constraint is fulfilled.

Conclusion and future works

In this article, a new strategy to ensure the dissipative constraint for a semi-active suspension is introduced, while keeping the advantages of the H control design. Interests of such an approach compared to existing ones are:

  • (1)

    Flexible design: possibility to apply H, H2, pole placement, mixed criteria, etc.

  • (2)

    Measurement: only the suspension deflection sensor (and its first derivative) is required.

  • (3)

    Computation: synthesis leads to two LTI controllers (of order 4) and a simple scheduling strategy

Acknowledgments

Authors would like to thank Pr. Ricardo Ramirez-Mendoza and Elvira Nino from the Tecnologico de Monterrey (Mexico) for their collaboration in the identification of the M-R damper model.

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