H-infinity Vehicle Control Using NonDimensional Perturbation Measures

Robust controller design techniques have been applied to the field of vehicle control to achieve many different performance measures: robust yaw rate control [1], robust lateral positioning using one [2, 3] or more [4-6] driver inputs, robust observer design, and so on. A difficulty with many published approaches is to obtain an adequate description of the model uncertainty. Most bounds on plant frequency responses or parameter perturbations are based on ad-hoc limits that rely primarily on the designer’s personal choice. The resulting controller design is therefore often vehicle specific, and is suitable only for application to a single design vehicle. This work shows that (a) the description of model uncertainty is not a design variable, (b) meaningful bounds on model uncertainty can be obtained from data and should be sought, and (c) these bounds can be used to generate a single controller suitable for any vehicle. 1. Previous Work and Problem Statement Previous work [3] presented a robust lateral-position controller useful for highway driving of any passenger vehicle. The system uncertainty was represented as matrix element perturbations of the system matrices. A Linear Matrix Inequality (LMI) approach was then used to design a state-feedback lateral position controller robust to expected parameter variation between vehicles. The resulting controller therefore robustly stabilizes all vehicles dynamically described by the bicycle model and which are parametrically bounded by fixed bounds. Unfortunately, these bounds do not address dynamic uncertainty associated with unmodeled dynamics, disturbances, or measurement noise. An important result of this previous work was the conclusion that a state-feedback controller is not capable of robust lateral vehicle positioning over wide variations in velocity, at least not without some type of gain scheduling. The controller presented in this work seeks to address controller robustness in a more intuitive framework than previous LMI representations. An H-infinity framework is presented that account for both parametric uncertainty and unmodeled dynamic uncertainty to achieve a robust controller design. A unique aspect of this work is the representation of vehicle dynamics in a nondimensional form that allows a generalized solution to the lateral control problem. Motivating this nondimensional representation is the desire to develop controller implementations suitable to any vehicle, not just a particular research vehicle under study. This paper is summarized as follows: Section 2 presents equations for the linear, lateral vehicle dynamics with a fixed preview distance. These equations are presented in both dimensional and nondimensional form, with the nondimensional form governed by a new set of unitless parameter groupings known as Π groups. Distributions of these Π groups define an average vehicle dynamic. Section 3 defines the robustness criteria required for generalized vehicle control: that a controller must stabilize the average vehicle dynamics in the presence of perturbations that generate the range of published vehicle dynamics. This range is numerically defined by bounding the frequency-response difference between the average vehicle and 50 other published vehicle dynamics. It was again found that no single controller is capable of robust control over large variations in speed or friction. Section 4 fixes the speed and friction, then develops a robust controller design to demonstrate a single-velocity robust controller implementation. Section 5 then discusses gainscheduling approaches and limitations to robust control. Finally, a conclusion summarizes the primary results. 2. Vehicle Dynamics Motivating the nondimensional representation is the desire to develop controller implementations suitable to any vehicle. Consequently, vehicle-to-vehicle variation is addressed in a nondimensional framework that utilizes the Buckingham Pi theorem [7]. The resulting representation accommodates two modeling aspects previously ignored by other researchers: first, parameter-to-parameter interdependency clearly arises due to common vehicle design; second, the individual parameter distributions show a normal distributions in the nondimensional representation, which specifically define a mean and standard deviation of vehicle dynamics [3, 8]. Application of the Buckingham Pi theorem [7] to the classical vehicle dynamics known as the Bicycle Model yields groupings of parameters that collectively do not have dimensions. These dimensionless parameters are known as Π groups, and for the planar bicycle model these parameters are: