1072 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 18, NO. 3, JUNE 2013 Adaptive Backstepping Control for Active Suspension Systems With Hard Constraints Weichao Sun, Member, IEEE, Huijun Gao, Senior Member, IEEE, and Okyay Kaynak, Fellow, IEEE Abstract—This paper proposes an adaptive backstepping con- trol strategy for vehicle active suspensions with hard constraints. An adaptive backstepping controller is designed to stabilize the at- titude of vehicle and meanwhile improve ride comfort in the pres- ence of parameter uncertainties, where suspension spaces, dynamic tire loads, and actuator saturations are considered as time-domain constraints. In addition to spring nonlinearity, the piecewise lin- ear behavior of the damper, which has different damping rates for compression and extension movements, is taken into consideration to form the basis of accurate control. Furthermore, a reference trajectory is planned to keep the vertical and pitch motions of car body to stabilize in predetermined time, which helps adjust acceler- ations accordingly to high or low levels for improving ride comfort. Finally, a design example is shown to illustrate the effectiveness of the proposed control law. Index Terms—Active suspension system, adaptive control, back- stepping control, hard constraints. NOMENCLATURE ˆ • Estimate of •. ˜ • Parameter estimation error of •. • max , • min Maximum and minimum values of •. • T Transpose of a matrix •. • > 0 (≥ 0) • is real symmetric and positive def- inite (semidefinite). λ min (•) and λ max (•) Minimal and maximal eigenvalues. ‖•‖ ∞ ∞-norm, which obeys ‖x‖ ∞ = max(|x j |),j =1,...,n. I. INTRODUCTION V EHICLE suspension systems are fundamental for sig- nificantly improving passenger comfort and handling characteristics. Roughly speaking, vehicle suspensions can be grouped into three types: passive, semiactive, and active Manuscript received December 7, 2011; revised May 10, 2012; accepted May 19, 2012. Date of publication July 10, 2012; date of current version January 18, 2013. Recommended by Technical Editor Y. Li. This work was supported in part by the 973 Project (2009CB320600), in part by the National Natural Science Foundation of China under Grant 60825303 and Grant 61021002, and in part by the Key Laboratory of Integrated Automation for the Process Industry (Northeastern University). W. Sun and H. Gao are with the Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, China (e-mail: 1984sunweichao@gmail.com; hjgao@hit.edu.cn). O. Kaynak is with the Department of Electrical and Electronic Engi- neering, Bogazici University, Istanbul 80815, Turkey (e-mail: okyay.kaynak@ boun.edu.tr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2012.2204765 suspensions. Passive suspensions, which comprise springs and dampers inserted between the body of vehicle and the wheel-axle assembly, are inadequate in improving ride comfort or road holding for the reason that these two criteria conflict with each other and necessitate variable spring and damper characteristics. Semiactive suspension systems with variable damping characteristics are also limited in improving ride comfort although they represent a considerable improvement over passive suspension systems [1]–[4]. Because appropriate improvements of active suspension sys- tems have a potential to improve the ride comfort and vehicle maneuverability, this research area has remained attractive for many years [5]–[9]. In active suspensions, actuators are placed between the car body and wheel-axle parallel to the suspension elements, and are able to both add and dissipate energy from the system, which enables the suspension to control the attitude of the vehicle, to reduce the effects of braking and the vehicle roll during cornering maneuvers to increase ride comfort and vehicle road handling. Since the actuators pull down or push up together with the suspension motions, the limitations arising from this should be taken into account. In order to make sure the car safety, the firm uninterrupted contact of wheels to road should be ensured, and the dynamic tire load should be small. In addition, actuator saturation may also appear in active suspen- sion control system. However, these design requirements are highly conflicting, for example, enhancing ride comfort calls for larger suspension stroke and smaller damping of wheel-hop mode and hence leads to a degradation in ride safety. In order to manage the tradeoff between conflicting require- ments, many active suspension control approaches are proposed, and a large number of different arrangements have been inves- tigated [10]–[13]. In classical control, pole location are used in [14] but all the nonlinear dynamics are neglected. In robust control, H ∞ and H 2 or the combination of both gives very sat- isfactory results, except for the fact that these studies are based on a linear model, while at high frequencies some unmodeled nonlinearities will rise up [15]–[18]. On the other hand, linear quadratic gaussian (LQG) approach is adopted in [19] to man- age the tradeoff between the ride comfort and control input, but here again the nonlinear dynamics are neglected, reducing hence the control efficiency. Regarding nonlinear control strate- gies, a sliding-mode controller is used in [20] and [21] and gives good closed-loop results. Another nonlinear controller that can be used is based on the backstepping technology as in [22]–[24], and the controller gives very good results, but performance re- quirements such as good road holding and actuator saturation are not considered. Compared with the existing results, most of which just con- sider partial performances, this paper proposes an adaptive 1083-4435/$31.00 © 2012 IEEE