In this paper, we present a novel control law for longitudinal speed control of autonomous vehicles. The key contributions of the proposed work include the design of a control law that reactively integrates the longitudinal surface gradient of road into its operation. In contrast to the existing works, we found that integrating the path gradient into the control framework improves the speed tracking efficacy. Since the control law is implemented over a shrinking domain scheme, it minimizes the integrated error by recomputing the control inputs at every discretized step and consequently provides less reaction time. This makes our control law suitable for motion planning frameworks that are operating at high frequencies. Furthermore, our work is implemented using a generalized vehicle model and can be easily extended to other classes of vehicles. The performance of gradient aware-shrinking domain based controller is implemented and tested on a stock electric vehicle on which a number of sensors are mounted. Results from the tests show the robustness of our control law for speed tracking on a terrain with varying gradient while also considering stringent time constraints imposed by the planning framework.
@cite_4: This paper presents a hybrid system approach to longitudinal speed and traction control of a vehicle with wheel slip constraints. The vehicle system is modeled as a hybrid system where the system is divided into local subsystems and each subsystem is selected to control the vehicle, in terms of control modes and operating region. Controllers are designed to track a desired speed value while maintaining the safety constraint that the absolute value of the slip between wheels and ground must be less than a given limit value to prevent the wheels from skidding or spinning. Simulation results are provided to show the feasibility of the proposed control system.
@cite_21: In this paper, we propose a novel reference model-based control approach for automotive longitudinal control. The reference model is nonlinear and provides dynamic solutions consistent with safety constraints and comfort specifications. The model is based on physical laws of compliant contact and has the particularity that its solutions can be explicitly described by integral curves. This allows to characterize the set of initial condition for which the constraints can be met. This model is combined with a simple feedback loop used to compensate unmodeled dynamics and external disturbances. Model simulations together with experimental results are also presented
@cite_16: 
@cite_10: This paper introduces the concept of a hybrid system and some of the challenges associated with the stability of such systems, including the issues of guaranteeing stability of switched stable systems and finding conditions for the existence of switched controllers for stabilizing switched unstable systems. In this endeavour, this paper surveys the major results in the (Lyapunov) stability of finite-dimensional hybrid systems and then discusses the stronger, more specialized results of switched linear (stable and unstable) systems. A section detailing how some of the results can be formulated as linear matrix inequalities is given. Stability analyses on the regulation of the angle of attack of an aircraft and on the PI control of a vehicle with an automatic transmission are given. Other examples are included to illustrate various results in this paper.
@cite_12: In the field of vehicle control, conventional cruise control systems have been available on the market for many years. During the last years, modern cars include more and more electronic systems. These systems are often governed by a computer or a network of computers programmed with powerful software. One of those new services is adaptive cruise control (ACC) (or autonomous intelligent cruise control, AICC), which extends the conventional cruise control system to include automated car following when the preceding car is driving at a lower speed than the desired set-speed. The focus of ACC has mainly been directed towards high-speed highway application, but to improve the comfort to the driver also low-speed situations must be considered. The paper presents an ACC system that is capable of car following in low-speed situations, e.g. in suburban areas, as well as in high-speed situations. The system is implemented in a test car and the result is evaluated.
In this section, we discuss the existing methods that have attempted to develop control laws for speed tracking. The initial work includes the development of control laws for vehicle following at both low speed and high speed scenarios @cite_12 . A non-linear vehicle system is modeled to obtain the dynamic solution with safety and comfort constraints in @cite_21 . In @cite_4 , @cite_16 , @cite_10 they even explored the hybrid system framework, where the vehicle system is divided into local subsystems.