On Performance Evaluation Methods and Control Strategies for Semi-Active Suspension Systems Sergio M. Savaresi, Enrico Silani, Sergio Bittanti Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza L. da Vinci, 32, 20133 Milano, ITALY. Nicola Porciani Ferrari S.p.A., Via Abetone inferiore, 4, 41053 Maranello (Modena), ITALY. ABSTRACT The problem considered in this paper is the design and analysis of a control system for semi-active suspensions in road vehicles. Three control strategies are tested: the two- state skyhook damping, the linear skyhook damping, and a newly-proposed control strategy based on two-state switching dampers. In order to assess and to compare the closed-loop performance of these control strategies, an evaluation method based on the describing-function is proposed. This evaluation approach provides, in the frequency-domain, a clear picture of the closed-loop performance on the body acceleration (comfort objective) and on the vertical load at the road-tire contact surface (handling objective). This evaluation method inherently provides a quantitative estimate of the degree of Input/Output non-linearity of the closed-loop system. 1. INTRODUCTION Among the many different types of controlled suspensions (load-leveling, adaptive shock absorbers, semi-active, slow-active, full-active, etc. – see e.g. [2-3, 6, 9-13, 15- 18]), semi-active suspensions have recently received a lot of attention, since they seem to provide the best compromise between costs (energy-consumption and actuators hardware) and performance. From one side, they only require shock-absorbers with controllable damping coefficient, which are known to be comparatively economic and to require negligible additional power. On the other hand, the main limit of semi-active suspensions is the “passivity-constraint”: it is impossible to deliver forces on the car body having the same direction of the suspension elongation speed. Semi-active suspensions are already mounted on many newly-designed top-cars, and a lot of academic and industrial research activity is currently undergoing in this area (see e.g. [3, 9, 11, 13, 18]). The innovative contribution of this paper is to propose a new control strategy, and to propose a new approach for the analysis of the closed-loop performance of semi-active suspension control systems. In order to assess and to compare the closed-loop performance of the control strategies, a frequency-domain evaluation method based on the describing-functions and on the notion of “variance gain” is here proposed. This evaluation approach – to the best of our knowledge - has never been applied to active and semi-active suspension systems. This evaluation method provides a clear picture of the closed-loop performance on the body acceleration (comfort objective) and on the vertical load at the road-tire contact surface (handling objective). An interesting feature of this evaluation method is that it inherently provides a quantitative estimate of the degree of Input/Output non- linearity of the closed-loop system. In order to better focus on the bulk of a semi-active suspension system, in this work we consider a quarter-car control problem only, since it represents the first stage for the development and testing of a full-body control system. The outline of this paper is the following: in Section 2 the problem is stated and a quarter-car system equipped with a controllable shock-absorber is briefly described and modeled; in Section 3 three different control strategies (two well-known strategies and a newly-proposed control strategy) are proposed. In Section 4 a performance- evaluation approach is described; this approach is used in Section 5 for the comparative evaluation of the three control strategies. 2. PROBLEM STATEMENT The dynamic model of a quarter-car system equipped with semi-active controlled suspensions is given by the following set of differential equations (see e.g. [12]): ( ) ( ) ( ) ( ) ( ) () () () () () () () () () () () () () () ( () ( )) () () () () 0 t t s t t t s t t r t t r t in Mzt ct zt z t k zt z t Mg mz t ct zt z t k zt z t k z t z t mg z t z t ct ct c t ct t β β  =− − − − −∆ −  =+ − + − −∆ +   − − −∆ −   − <∆   =− + ≥ ∀  $$ $ $ $$ $ $ $ (1) where (see Fig.1): ( ), ( ), () t r zt z t z r are the absolute vertical positions of the body, the unsprung mass, and the road profile, respectively; M is the quarter-car body mass; m is the unsprung mass (tire, wheel, brakes, suspension links, etc.); c(t) and () in c t are the actual and the requested damping coefficients of the shock-absorber, respectively; note that c(t) is subject to the “passivity-constraint”: () 0 ct ≥ ; β is the bandwidth of the active shock-absorber; k and k t are the stiffness of the suspension spring and of the tire, respectively; s ∆ and t ∆ are the length of the unloaded suspension spring and tire, respectively; g is the Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003 WeM08-4 0-7803-7924-1/03/$17.00 ©2003 IEEE 2264