416 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 19, NO. 2, MARCH 2011 Finite Frequency Control for Vehicle Active Suspension Systems Weichao Sun, Huijun Gao, Senior Member, IEEE, and Okyay Kaynak, Fellow, IEEE Abstract—This brief addresses the problem of control for active vehicle suspension systems in finite frequency domain. The performance is used to measure ride comfort so that more general road disturbances can be considered. By using the gener- alized Kalman–Yakubovich–Popov (KYP) lemma, the norm from the disturbance to the controlled output is decreased in spe- cific frequency band to improve the ride comfort. Compared with the entire frequency approach, the finite frequency approach sup- presses the vibration more effectively for the concerned frequency range. In addition, the time-domain constraints, which represent performance requirements for vehicle suspensions, are guaranteed in the controller design. A state feedback controller is designed in the framework of linear matrix inequality (LMI) optimization. A quarter-car model with active suspension system is considered in this brief and a numerical example is employed to illustrate the ef- fectiveness of the proposed approach. Index Terms—Active suspension systems, constraints, finite fre- quency, generalized KYP lemma, control. I. INTRODUCTION A VEHICLE suspension system basically consists of wish- bone, spring, and shock absorber to transmit and filter all forces between body and road. The spring is to carry the body-mass and to isolate the body from road disturbances and thus contributes to ride comfort. The task of the damper is the damping of body and wheel oscillations, where the avoidance of wheel oscillations directly refers to ride safety. Since the vehicle suspension system is responsible for ride comfort and safety, it plays an important role in modern vehicles. In recent years, a lot of efforts have been made to develop models for suspension systems and to define design specifica- tions that reflect the main objectives to be taken into account. In this sense, ride comfort, road-holding ability, suspension de- flection, and actuator saturation are important factors to be ad- dressed by any control scheme. However, these requirements are conflicting. For example, increasing ride comfort results in larger suspension stroke and smaller damping in the wheel-hop mode. Therefore, vehicle suspension design requires a com- promise between ride comfort and vehicle control. To achieve Manuscript received July 24, 2009; revised January 09, 2010. Manuscript received in final form January 28, 2010. First published March 11, 2010; cur- rent version published February 23, 2011. Recommended by Associate Editor C.-Y. Su. This work was supported in part by National Outstanding Youth Sci- ence Fund (60825303), by 973 Project (2009CB320600), by the Heilongjiang Outstanding Youth Science Fund (JC200809), and by the Foundation for the Author of National Excellent Doctoral Dissertation of China (2007B4). W. Sun and H. Gao are with the Space Control and Inertial Technology Re- search Center, Harbin Institute of Technology, Harbin 150001, China (e-mail: 1984sunweichao@gmail.com; hjgao@hit.edu.cn). O. Kaynak is with the Electrical and Electronic Engineering De- partment, Bogazici University, Bebek 80815, Istanbul, Turkey (e-mail: okyay.kaynak@boun.edu.tr). Color versions of one or more of the figures in this brief are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2010.2042296 a compromise between the performance requirements, a con- siderable amount of research has been carried out for the last few decades [3], [17], [21]. Among the proposed solutions, ac- tive suspension is a possible way to improve suspension per- formance and has attracted much attention [10], [19], [24], and many active suspension control approaches are proposed, based on various control techniques such as linear quadratic Gaussian (LQG) control [4], adaptive control and nonlinear control [11], fuzzy logic and neural network control [15], and control [14]. In particular, active suspensions have been intensively discussed in the context of robustness and disturbance attenua- tion [6], [7]. Therefore, in recent years, more and more attention has been devoted to the control of active suspensions, and a number of important results have been reported, see for ex- ample, [5], [13] and the references therein. The most important objective for vehicle suspension systems is the improvement of ride comfort. In other words, the main task is to design the controller which can succeed in stabilizing the vertical motion of the car body and isolating the force trans- mitted to the passengers as well. In the literature it is possible to find many results which aim at improving ride comfort [8], [20], [22]. These results can effectively achieve desired vehicle suspension performance, especially the ride comfort. It is worth mentioning that most of the reported approaches are considered in the entire frequency domain. However, active suspension sys- tems may just belong to certain frequency band, and ride com- fort is known to be frequency sensitive. From the ISO2361, the human body is much sensitive to vibrations of 4–8 Hz in the ver- tical direction. Hence, the development of control in finite frequency domain is significative for active suspension systems. The current approach for finite frequency domain is to introduce the weighting functions. The weighting method is useful in practice, however, the additional weights increase the system complexity. Besides, the process of selecting appro- priate weights is time-consuming, especially when the designer has to shoot for a good tradeoff between the complexity of the weights and the accuracy in capturing desired specifications. An alternative approach is to grid the frequency axis. This ap- proach has a practical significance especially when the system is well damped and the frequency response is expected to be smooth. But it lacks a rigorous performance guarantee in the design process. Another approach that avoids both weighting functions and frequency gridding is to generalize the fundamental ma- chinery, the Kalman–Yakuboviˇ c–Popov (KYP) lemma. The KYP lemma establishes the equivalence between a frequency domain inequality for a transfer function and a linear matrix inequality (LMI) associated with its state-space realization [1], [9], [12]. It allows us to characterize various properties of dynamic systems in the frequency domain in terms of LMIs. However, the standard KYP lemma is only applicable for the infinite frequency range. Recently, a very significant devel- 1063-6536/$26.00 © 2010 IEEE