Decentralized Robust H∞ Controller Design for a Half-car Active Suspension System

In this paper an H∞ controller is designed for a hydraulically actuated active suspension system of a half-modeled vehicle in a cascade feedback structure. Using the proposed structure the nonlinear behavior of actuator is reduced significantly. In the controller synthesis, a proportional controller is used in the inner loop, and a robust H∞ controller forms the outer loop. Two H∞ controllers are designed for this system. First unstructured uncertainty is not considered in the design procedure and secondly, the controller is designed considering uncertainty. Each of these controllers is designed in a decentralized fashion and the vehicle oscillation in the human sensitivity frequency range is reduced to a minimum. Statistical analysis of the simulation result using random input as road roughness, illustrates the effectiveness of the proposed control algorithm for both cases. Keyword: active suspension, cascade feedback, hydraulic actuator, nonlinear model, multiplicative uncertainty, robust H∞ controller, disturbance input, road random input, statistical analysis. INTRODUCTION Demands for ride comfort and controllability of road vehicle are pursued by many automotive industries by using active suspension. These electronically controlled suspension system can improve the ride comfort as well as the road handling of the vehicle. The suspension system can be categorized into three groups: passive suspension systems including conventional springs and dampers. These systems contain no electronic sensor and control, Miller (1). Semi-active suspension systems provide controlled real-time dissipation of energy, (1). Active suspension systems use a hydraulic or pneumatic actuator in parallel with a passive spring and shock absorber, and hence, the measurement of body vibration is used to decide instantaneously the amount of force needed by the actuator. Different characteristics can be considered in a suspension system design namely, ride comfort, body movement, road holding and suspension travel. No suspension system can simultaneously optimize all four mentioned parameters. But a better trade-off among these parameters can be achieved in active suspension system, Taghirad and Esmailzadeh (2). Many researches in recent years have concentrated on active vehicle suspension system. In these researches a variety of models including 1/4, 1/2 and full-car model have considered. Although experiments show the importance of nonlinear behavior of actuator in determination of a suitable trade-off in a suspension system, in many published researches, no attention is made to the nonlinear behavior of this element, (2), Yamashita et al (3), Wang et al (4). In Some researches use backstepping control is used to investigate nonlinear behavior of actuator in different road condition, Lin and Kanellakopoulos ( 5), Karlsson et al ( 6). Moreover, nonlinear optimal control is proposed in Karlsson et al (7, 8), which provides a controller for active suspension system. Nonlinear H∞ control is another design approach for a quarter-car model, Karlsson et al (9). Also, use of cascade feedback structure is another way for investigation of hydraulic actuator nonlinearity to reduce nonlinear behavior of hydraulic actuator, Fukao et al ( 10). In this paper a decentralized robust H∞ control are designed for a half-modeled vehicle considering nonlinear behavior of hydraulic actuator. Nonlinear system is linearized in different operating conditions and linearization results show the dominance of nonlinear behavior of the hydraulic actuator. To remedy this drawback, cascade feedback structure is used. Using this structure, the behavior of the system is significantly linearized makes it plausible to determine linear model for the system in addition to minimum bounded norm multiplicative uncertainty description. Two H∞ controllers are designed for the above structure, considering nominal performance for the first case, and robust performance in the second. The solution of mixed sensitivity problem is significantly reducing the vehicle vibration in the human sensitivity frequency range. Statistical analysis of the simulation results using random input as road roughness illustrates that the proposed strategy can provide a suitable trade-off between ride comfort and road holding, despite nonlinear behavior of the actuator. HALF – CAR MODEL Figure 1 illustrates the half–car model of a passenger car, in which only four degrees of freedom are considered, Taghirad and Behravesh (15). In this model the dynamical motion of the vehicle body and two axles in longitudinal plane is determined. The suspension stiffness and the tires are modeled by linear spring in parallel with viscous dampers. Force-generating Control 2004, University of Bath, UK, September 2004 ID-216 elements in active suspension system are hydraulic actuators, Alleyn and Hedrick (13). The systems with four degrees of freedom are represented by the following states: body bounce Z s , body pitchθ , front and rear tire deflection, z u1, z u2. Figure 1 : Half-car Suspension System The following equations of motion are derived for the model using Newton – Euler method: ) ) ( ) ( ) ( ) ( ( 1 2 1 2 1 2 2 1 1 1 2 2 1 1 1 2 1 1 1 1 1 f f A A s s s s s u s u uz uz sz u s s s s s s s s f F F F b B B z B B z B z B z K z K b K b K z K K M Z − − + + − − + − + + + + − − + − = θ θ & & & & & & ) ( ) ( ) ( ) ( ) ( (( 1 2 2 1 1 2 2 2 1 1 1 2 2 1 2 1 1 2 2 1 1 2 2 2 1 1 1 2 2 1 2 1 1 2 1 1 1 b F b F z b B z b B b B b B z b B b B z b K z b K b K b K z b K b K I A A u s u s s s s s s u s u s s s s s s − + + − + − − − + − + − − − =