Proceedings of COBEM 2005 18th International Congress of Mechanical Engineering Copyright © 2005 by ABCM November 6-11, 2005, Ouro Preto, MG FOUR-WHEEL VEHICLE SUSPENSION MODELING FOR CONTROL SYSTEM DEVELOPMENT Cláudio Crivellaro DANA Structural Solutions International – Brazil Division (Dana Indústrias Ltda) Av. Presidente Médici, 939, Zip code 06268-000, Osasco, SP, Brazil e-mail: claudio.crivellaro@dana.com Edilson Hiroshi Tamai State University of São Paulo (USP) – Escola Politécnica – DEM Zip code 05508-900, São Paulo, SP, Brazil e-mail: edhtamai@usp.br Décio Crisol Donha State University of São Paulo (USP) – Escola Politécnica – DEM Zip code 05508-900, São Paulo, SP, Brazil e-mail: decdonha@usp.br Abstract. In recent years several works have been published regarding active and semi-active suspension system for vehicles in general; however few articles have dealt with an adequate four-wheel suspension model for control development. When a four-wheel model is used, several control problems may appear, such as the presence of non- minimum phase transmissions zeros, the lack of observability and controllability, and ill-conditioned systems. In this work a seven-degree of freedom four-wheel vehicle model is presented, and a balanced state-space realization and model reduction is proposed to get a fully controllable and observable model for vehicle suspension controller design. In addition a robust control (LQG/LTR) is designed and simulations results comparing passive and active systems are presented. Keywords: Vehicle suspension, four-wheel model, balanced realization, space-state, robust control. 1. Introduction The vehicle mathematical modeling is the base of several control strategies applied to vehicle suspension systems. In recent years several works have been published regarding active and semi-active suspension system for numerical simulations. Some works dealing with different kind of control approach used just a quarter-car model with two degree of freedom (Rao and Prahlad, 1995; Stutz and Rochinha, 2005). Other works aimed at other vehicles movements such as roll or pitch rather then with the control approach have presented half-car models with four degree of freedom (Ha et al., 1996; Tsao and Chen, 2001; Simon and Ahmadian, 2002). Other works such as Cruz et al., 2003, and Yamamura and Masada, 1979, have used four-wheel modeling for MAGLEV vehicle controller development, but did not use a MIMO (multiple input multiple output) control approach, although Cruz et al., 2003, had to deal with the case of an over actuated system, solved with an optimal distribution of forces among the actuators designed to minimize the maximum force. The goal of this work is to propose an alternative strategy of vehicle modeling, representative of a full four-wheeled vehicle, which is suitable for MIMO control development. The MIMO control approach can consider the global performance of vehicle movements to determine the best control action, which is, a priori, a better choice than the use of several SISO local controllers working independently. In addition, a state-space theory is an elegant way to approach a control problem, mainly regarding MIMO systems which are naturally dealt. This theory has given important concepts such as observability and controllability, and leaded to several control design methods, such as linear quadratic regulator (LQR), pole placement, optimal H 2 , and robust control design methods as LQG-LTR (Doyle and Stein, 1981; Cruz, 1996) and H  (Glover and Doyle, 1989). The choice of MIMO approach for suspension system control development brings up several control issues as for example the presence of non-minimum phase transmission zeros. A right half plane zero gives an upper bound to the achievable bandwidth. The bandwidth decreases with decreasing frequency of the zero. It is thus more difficult to control systems with slow zeros. On the other hand, when the vehicle body is modeled as a rigid body, the four independent suspension system (one for each wheel) can only control three (heave, pitch and roll) from the six degree of freedom. In this case, it is very common to use four actuators – one for each wheel – and just three linear-independent signals could be measured to determine uniquely the position of vehicle body (heave, roll and pitch movements) or its acceleration state (heave, roll and pitch accelerations). In this way, a four dimensional control signal cannot influence some movements of vehicle body, which cannot be represented in a three dimensional space, for example, when the control signals excite a twist mode of car body structure. This situation is extended to the system output when, for example, the vertical acceleration