H Infinity Lateral Control Of Autonomous Vehicles

H Infinity Lateral Control Of Autonomous Vehicles

Author: SHAHBAZ ALI

Abstract

This report delves into the development of lateral control for autonomous vehicles using an H∞ controller. The goal is to create a system that can effectively guide a vehicle along a desired trajectory. The modeling of vehicle dynamics employs a linear bicycle model, emphasizing lateral movements. The H∞ controller is chosen for its robustness in handling uncertainties in unpredictable environments.

The design process involves loop shaping using weight functions, and MATLAB is utilized for implementation and performance analysis. Simulations in Simulink showcase the controller's ability to track a reference trajectory. A trajectory generation method, incorporating curvature calculations, tests the controller under realistic conditions. Results demonstrate effective tracking, though further robustness analysis and optimization are recommended for future enhancements.

Introduction

Autonomous Driving applications have become realizable and feasible due to the advancements in various fields such as machine learning and computational systems. Pairing this with the advancements in hardware, we can now establish an advanced complex system to drive a car autonomously.

For the last 25 years robotic vehicles, which are commonly known as intelligent vehicles, have become a key factor in robotics and in autonomous vehicles. The goal is to be able in the next 20-30 years to achieve a fully driving autonomous car experience, but there are still some challenges ahead that need to be overcome. Some of those challenges are traffic laws and safety regulations. Achieving such a car will aid us in reducing the number of accidents, air pollution, and noise pollution.

Today, it is believed that there are over one billion passenger vehicles traveling on the highways around the world. Sometimes with huge traffic, there are many scenarios where the driver cannot make a fast decision, in most cases, the driver cannot react very quickly which is why automated systems will do the various tasks for them, which can perform different tasks that the driver might encounter while operating a vehicle.

An intelligent vehicle is explained as a vehicle that is supplied with perceptual tools that is allowed to do some automation tasks which are for instance, avoidance of obstacles, overcoming slow traffic, and following the safe lane. The inspiration for the development of autonomous intelligent vehicles is focused on adaptation, efficiency, and the safeness of the roads. Active Safety (AS) and Passive Safety (PS) are in the scope of interest and complement each other. Active safety improves vehicle stability and controllability when facing dangerous, and emergency situations. Passive safety is concerned with the integrity and confidentiality of the vehicle .Currently, there is a lot of industrial research and development ongoing in the design of the control of autonomous vehicles[1].

Objective

With the demand of autonomous vehicles due to its advantages and comfort. It is necessary to implement a control to safely drive it and not let dangerous situations cause an accident. This projects aims to create a lateral control for autonomous vehicle that can direct the vehicle to a desired trajectory

Modelling of Vehicle Dynamics

The modeling of vehicle dynamics is taken as a linear bicycle model as proposed by authors in [2]. The movement of the vehicle is explained using a bicycle model in which the longitudinal (front and back) and lateral (side-to-side) movements of the vehicle are separated. For designing the controller, a simplified linear model is used to describe the lateral dynamics, focusing on the side-to-side and yaw (rotation) movements. Front two wheels and rear two wheel pairs are coupled together and hence the name bicycle model as we consider a two wheel model as given in figure 1.

Figure 1: Bicycle Model of Vehicle Dynamics

Where the parameters are given below:

v	Linear velocity vector (vx , vy), vx denotes speed
α_f ; α_r	Side slip angles of the front and rear tires vehicle
ψ	Yaw angle within a fixed inertial frame
δ_f	Front wheel steering angle
δ	Commanded steering angle
m	Total mass of the vehicle
Iψ	total inertia vehicle around center of gravity (CG)
l_f ; l_r	distance of the front and rear axles from the CG
l	distance between the front and the rear axle l_f + l_r
c_f ; c_r	cornering stiffness of the front and rear tires

Lateral Dynamics

Using the Newton-Euler method in [3] and choosing vy and ψ as state variable and an steering wheel angle as an input, we get the space state representation of our vehicle dynamics as:

Where

, , , , ,

H∞ Control Design

Why H∞ Control?

Autonomous vehicles use the H-infinity controller to solve a number of important problems related to control system design in unpredictable and complicated situations. Robust control theory is the foundation of the H∞ control methodology, which works especially effectively in applications with lots of uncertainties, disturbances, and changing operating conditions.

H∞ control has plenty of key factors in autonomous vehicles but some of the reasons are for instance, Autonomous vehicles work in constantly changing and unpredictable surroundings, wherein variables like weather, road conditions, and inaccurate sensor readings add unpredictability to the system.The H∞ controller reduces the influence of uncertainties on the control system in order to deliver reliable performance. It accomplishes this by maximizing the performance of the closed-loop system under extreme circumstances.

Rejecting external disturbances effectively is a critical function of H∞ controllers, as it preserves the stability and functionality of autonomous vehicles. Road imperfections, wind, and other disturbances can all have a big impact on the dynamics of the car. The closed-loop system's susceptibility to these disturbances is intended to be reduced by the H∞ controller.

Autonomous cars frequently need to strike a compromise between multiple performance goals, including tracking precision, energy efficiency, and stability. The systematic trade-off between these competing goals is made possible by the H∞ control framework. To reach a desired level of performance and guarantee robustness against uncertainties, design parameters might be changed.

Because the H∞ control methodology acts in the frequency domain, system dynamics can be taken into account over a wide range of frequencies. Autonomous cars frequently come with uncertainty and disruptions with varying frequency characteristics. Because of the H∞ controller's skill in handling these variances, overall performance is enhanced.

Designing Controller

The design of H∞ controller for the lateral control of vehicles is motivated by the work of Hussam Atoui[4]. As it is commonly understood, the design of a robust controller aims to minimize the H∞ norm of the plant, thereby optimizing its performance. To achieve this objective, three weight functions are introduced into the plant for loop shaping. These weight functions, functioning as lead-lag compensators, have the capability to adjust the frequency response of the system according to specified requirements. The process of loop shaping, utilizing these weight functions, is employed to attain the desired frequency response for the plant.

Various methodologies exist for loop shaping, and the parameters of the weight functions are adjusted to ensure that the frequency response of the entire system conforms to predetermined specifications. The mixed sensitivity problem is illustrated in the block diagram depicted in Figure 2, providing a visual representation of the considerations involved in achieving the desired system performance.

Figure 2: Model for the design of Hinf controller

Figure 3: General Control Problem

The weighting function templates are selected as follows:

Template We specification on tracking

Where the parameters are tuned as follows:

Ms = 2, to ensure robustness at any frequency.
wb ≥ 10, to choose the rate of minimization.
Ɛ ≤ 10^-¹, to represent the steady-state tracking error.

Template Wu specification on control input limitations

The parameters are adopted as:

Mu represents the limitations on the maximum allowed gain.
wbu, is related to the control bandwidth.
≤ 10^-², is concerned with the noise rejection at high frequencies.

MATLAB Implementation

Firstly the weighting functions with its parameters are defined in the MATLAB as follows:

Figure 4: Weighting template parameters

For the mixed sensitivity H∞ control problem the Matlab is used to get the generalized

plant P using sysic command as in the following code:

Figure 5: Finding H infinity controller

sysic will perform the interconnection described by the variables defined and create a generalized plant P that is used to find optimal Hinfinity controller using hinfsyn. The hinfsyn function is finds a controller (khinf1) that minimizes the H-infinity norm of the closed-loop transfer function, subject to specified performance and stability constraints.

H∞ Performance Analysis

The sensitivity functions provide insights into how the closed-loop system responds to changes in various inputs, such as reference signals, disturbances, and noise. The analysis includes computing and visualizing sensitivity functions for different aspects of the closed-loop system. We plotted sensitivity functions using sigma in MATLAB and results are given below:

Figure 6: Plot of sensitivity functions

Analysis of S:

At low frequency < -40dB, low steady state error
At high frequency < 6dB, good modulus margin
Good rejection of output disturbance

Analysis of T:

At low frequency, good tracking |T| = 0dB
At high frequency, good noise rejection, < 50dB

Analysis of SG:

At low frequency < 25dB, good rejection of input disturbance

Analysis of KS

Less than template, no input saturation
At high frequency, < 35dB, rejection of noise

Simulation in Simulink

Following figure 7 gives the configuration of the scheme. Where khinf1 is the controller state space and G is the vehicle model state space. Input data of reference is a CSV database having yaw rate reference pertaining to a desired trajectory.

Figure 7: Control Scheme in Simulink

As it can be seen from the figure 8, the output is plotted against reference. The output response is very closely tracking to the reference with small steady state error.

Figure 8: Plot of output yaw rate vs reference yaw rate

Trajectory Generation

To be able to test our controller and evaluate its performance for a given trajectory, we have used the following method to generate the trajectory and test it on MATLAB.

Equations:

Where:

t is the parameter representing time
x(t) and y(t) and the coordinates of the trajectory.

Curvature Calculation

After running the specific script for generating we have obtained the following trajectory:

A graph with a line

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Figure 9: Trajectory Obtained

The curvature of the road at any point (x,y), is calculated using the following formula:

Curvature =

Where:

is the tangent angle of the trajectory.
is the arc length calculated as .
The tangent angle is given by .

Implementation and Results

After obtaining the curvature data, the data was then stored in an excel file used as a control signal input for the controller. After running the simulations, we have obtained the following results: Steering angle

A table of numbers with a white background

Description automatically generated A graph of a function

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Figure 10: Data obtained Figure 11: Control Signal

Figure 12: Reference vs Controlled

Conclusion on above result

As we can see from figure 12, our controller is doing a decent job in tracking the reference element. It can be noticed at time= 0 that there is some sort of delay till when the controller starts working, this could be explained by the fact that the controller is working at time = 0 where our road is still not reached. With those results a robustness analysis must be made to furthermore analyze the results. Also more optimization must be made, where the trajectory data should be directly collected from actual driving.

Conclusion

In conclusion, this report has explored the development and implementation of lateral control for autonomous vehicles, employing an H∞ controller. The significance of autonomous driving applications and the challenges posed by safety regulations and traffic laws were acknowledged. The project successfully aimed at creating a lateral control system focused on trajectory tracking.

The chosen linear bicycle model for vehicle dynamics and the H∞ controller have proven effective in handling uncertainties and disturbances inherent in the complex and dynamic environments in which autonomous vehicles operate. The simulations demonstrated the controller's ability to closely track a reference trajectory, showcasing its potential for real-world applications.

However, the report acknowledges certain limitations and areas for improvement. The shift in project goals due to hardware limitations impacted the original vision of incorporating vision dynamics for a lookahead system. The observed delay in controller response at the initial time step requires further investigation and optimization. Robustness analysis is recommended to ensure the controller's performance in diverse scenarios.

Furthermore, future work could focus on direct data collection from actual driving scenarios to enhance trajectory data accuracy. Additionally, ongoing optimization efforts should aim at refining the system's response to various operating conditions.

In essence, while the current lateral control system exhibits promising results, continuous refinement and adaptation will be essential for meeting the evolving demands of autonomous vehicle technology and ensuring reliable and safe operation in real-world conditions.

References

[1] M. Ammour, R. Orjuela, and M. Basset, “Collision avoidance for autonomous vehicle using MPC and time varying Sigmoid safety constraints,” IFAC-PapersOnLine, vol. 54, no. 10, pp. 39–44, 2021, doi: https://doi.org/10.1016/j.ifacol.2021.10.138.

[2] https://medium.com/@viraj.killedar20/case-study-on-the-implementation-of-the-autonomous -driving-4459741877cc.

[3] Guo, J., Hu, P., & Wang, R. (2016). Nonlinear coordinated steering and braking control of vision-based autonomous vehicles in emergency obstacle avoidance. IEEE Transactions on Intelligent Transportation Systems, 17(11), 3230-3240.

[4] Atoui, Hussam, et al. "Real-time look-ahead distance optimization for smooth and robust steering control of autonomous vehicles." 2021 29th Mediterranean Conference on Control and Automation (MED). IEEE, 2021.

[5] X. He, Y. Liu, C. Lv, X. Ji, and Y. Liu, “Emergency steering control of autonomous vehicle for collision avoidance and stabilisation,” Vehicle System Dynamics, vol. 57, no. 8, pp. 1163–1187, Nov. 2018, doi: https://doi.org/10.1080/00423114.2018.1537494

[6] Poussot-Vassal, C., O. Sename, L. Dugard and S. M. Savaresi Vehicle dynamic stability improvements through gainscheduled steering and braking control. . Vehicle System Dynamics, (2011). 49:10, : p. 1597–1621.

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H Infinity Lateral Control Of Autonomous Vehicles