Vehicle Dynamics Control Using Control Allocation
Karan Chatrath Master of Science Thesis
Department Of Cognitive Robotics (CoR)
Vehicle Dynamics Control Using Control Allocation
Master of Science Thesis
For the degree of Master of Science in Vehicle Engineering at Delft University of Technology
Karan Chatrath
July 12, 2019
Faculty of Mechanical, Maritime and Materials Engineering (3mE) · Delft University of Technology Copyright c Department Of Cognitive Robotics (CoR) All rights reserved. Delft University of Technology Department of Department Of Cognitive Robotics (CoR)
The undersigned hereby certify that they have read and recommend to the Faculty of Mechanical, Maritime and Materials Engineering (3mE) for acceptance a thesis entitled Vehicle Dynamics Control Using Control Allocation by Karan Chatrath in partial fulfillment of the requirements for the degree of Master of Science Vehicle Engineering
Dated: July 12, 2019
Supervisor(s): Dr. Barys Shyrokau
Yanggu Zheng
Reader(s): Dr.ir.Tamas Keviczky
Dr. Bilge Atasoy
Abstract
Advancement of the state of the art of automotive technologies is a continuous process. It is essential for automotive engineers to combine the knowledge of vehicle dynamics and control theory to develop useful applications that meet requirements of improved safety, comfort and performance. A road vehicle is equipped with several actuators that can assist a user during a dynamic driving task and ensure overall system reliability. Using all available actuators effectively to make a vehicle move in the desired manner is necessary. Typically, the available actuators outnumber the states of motion to be controlled. Such mechanical systems are referred to as over-actuated.
An effective way to control an over-actuated system is through the use of control allocation (CA).CA ensures coordination between, and the optimal use, of all available actuators. This strategy also considers the limits of the actuators. Despite its features, a lot ofCA methods have a drawback that actuator dynamics are neglected. This drawback has been addressed with a method called model predictive control allocation (MPCA). The behaviour of mechanical actuators is usually approximated by simplified models. Un-modelled system dynamics are always a source of uncertainty. Also, the aging of actuators introduces the element of uncertainty. The ability of MPCA to handle uncertainties is investigated and a solution is proposed to overcome this shortcoming. The proposed solution is the combination of an online adaptive parameter estimator with the MPCA strategy. This way, the CA solver is constantly updated with the parameters of each actuator. This technique is used to design vehicle stability controllers and their performance on simulation is reported.
The results indicate that the proposed control allocation technique is effective for vehicle stability control in various scenarios. However, scope for betterment has been recognised and relevant recommendations are made, to conclude this work.
Master of Science Thesis Karan Chatrath ii
Karan Chatrath Master of Science Thesis Table of Contents
Acknowledgements xiii
1 Introduction1 1-1 Introduction to Electronic Stability Control...... 2 1-2 Over-actuated Mechanical Systems...... 3 1-3 Control Allocation...... 4 1-4 Problem Definition...... 5 1-5 Summary Of Work Done...... 5 1-6 Contributions...... 7 1-7 Layout of This Master Thesis...... 7
2 Vehicle Dynamics Modelling9 2-1 IPG CarMaker Multi Body Vehicle Model...... 9 2-2 Planar Vehicle Model...... 10 2-2-1 Vehicle Body...... 10 2-2-2 Wheel and Tire Related Quantities...... 13 2-3 Tire Modelling...... 14 2-3-1 Linear Tire Model And Friction Circle...... 14 2-3-2 Dugoff Tire Model...... 15 2-4 The Linear Bicycle Model...... 15 2-5 Actuator Dynamics...... 16 2-5-1 Brake Actuator Dynamics...... 16 2-5-2 Steering Actuator Dynamics...... 18 2-6 Dynamic Driving Manoeuvre...... 19 2-7 Validation Of The Planar Model...... 20 2-8 Summary...... 22
Master of Science Thesis Karan Chatrath iv Table of Contents
3 Control Allocation Theory 23 3-1 Control Allocation Problem Formulation...... 23 3-2 Control Allocation Methods - A Brief Literature Review...... 24 3-3 Weighted Least Squares Control Allocation...... 26 3-4 Dealing With Actuator Dynamics...... 27 3-5 Model Predictive Control Allocation...... 28 3-6 A Simple Example...... 31 3-7 MPCA With Adaptive Parameter Estimation (APE)...... 34 3-7-1 Online adaptive parameter estimation...... 35 3-7-2 Combining APE with MPCA...... 36 3-8 Summary...... 38
4 Electronic Stability Control Using Control Allocation 39 4-1 Sine With Dwell Test With No Control...... 39 4-2 General Layout Of The Stability Control System...... 40 4-3 Reference Generator...... 41 4-4 High-Level Controller...... 43 4-5 Control Effectiveness Matrix Derivation...... 44 4-5-1 Case 1: 4 Actuators - Differential Braking...... 45 4-5-2 Case 2: 5 Actuators - Differential Braking And Active Front Steering.. 45 4-6 Summary Of All Simulation Scenarios...... 46 4-6-1 Configuration 1: With Four Actuators...... 46 4-6-2 Configuration 2: With Five Actuators...... 46 4-6-3 Additional Scenarios...... 47 4-7 Details Of Simulation Scenarios and Results...... 47 4-7-1 Tuning Parameters...... 48 4-7-2 Simulation Scenario Case E...... 51 4-7-3 Simulation Scenario - Case J...... 53 4-8 Additional Vehicle Dynamics Factors...... 54 4-8-1 Tire Limits As Control Allocation Constraints...... 55 4-8-2 Variation of Cornering Stiffness...... 56 4-9 Simulations Scenarios: Cases K and M...... 58 4-9-1 Simulations With Hydraulic Brake Model...... 60 4-10 Summary...... 63
5 Conclusions And Recommendations 65 5-1 Highlights And Conclusions...... 65 5-2 Recommendations And Scope For Future Work...... 66
A Planar vehicle model and Dugoff model validation 69
Karan Chatrath Master of Science Thesis Table of Contents v
B ESC with Control Allocation - Results 73 B-1 Case A Results...... 73 B-2 Case B Results...... 75 B-3 Case C Results...... 77 B-4 Case D Results...... 79 B-5 Case E Results...... 80 B-6 Case F Results...... 81 B-7 Case G Results...... 83 B-8 Case H Results...... 85 B-9 Case I Results...... 87 B-10 Case J Results...... 88 B-11 Case K Results...... 91 B-12 Case L Results...... 93 B-13 Case M Results...... 95
Bibliography 97
Glossary 101 List of Acronyms...... 101
Master of Science Thesis Karan Chatrath vi Table of Contents
Karan Chatrath Master of Science Thesis List of Figures
1-1 Working Of ESC Using Differential Braking...... 2 1-2 Control allocation general strategy...... 4
2-1 Planar vehicle motion indicating coordinate frames...... 10 2-2 Hydraulic brake model schematic for a single wheel...... 17 2-3 Pfeffer Steering Model...... 19 2-4 Sine With Dwell Test Steering Input...... 20 2-5 Validation with Steering wheel angle amplitude of 120 degrees...... 21 2-6 Dugoff Tire Model Validation - Steering Wheel Angle 120 degrees...... 21
3-1 Weighted Least squares CA block diagram...... 27 3-2 Control allocation with actuator dynamics considered...... 28 3-3 Simple example - WLS - CA - No actuator Dynamics...... 32 3-4 Simple example - WLS - CA - With actuator Dynamics...... 33 3-5 Simple example - MPCA...... 33 3-6 Simple example - MPCA with actuator uncertainties...... 34 3-7 Simple example - APE with MPCA...... 37 3-8 Simple example - APE with MPCA - Parameter Convergence...... 37
4-1 Vehicle Response With No Control...... 40 4-2 General Block Diagram for ESC Using Control Allocation...... 41 4-3 Reference signals for yaw rate control for the sine with dwell test...... 42 4-4 MPCA BLOCK Diagram - General...... 49 4-5 APE+MPCA BLOCK Diagram - General...... 50 4-6 Sine With Dwell - Case E - Vehicle Response...... 52 4-7 Sine With Dwell - Case E - Actuator Response...... 52
Master of Science Thesis Karan Chatrath viii List of Figures
4-8 Sine With Dwell - Case E - SWA - 130 degrees...... 53 4-9 Variation of Cornering stiffness with time during the sine with dwell manoeuvre. 57 4-10 APE with MPCA accounting for tire limits and dynamic control effectiveness.. 58 4-11 Sine With Dwell Test - Case K - Vehicle Response...... 58 4-12 Sine With Dwell Test - Case K - Actuator Response...... 59 4-13 Sine With Dwell Test - Case J and K Comparison...... 60 4-14 Sine With Dwell Test - Case M - Vehicle Response...... 61 4-15 Sine With Dwell Test - Case M - Vehicle Response...... 61 4-16 Sine With Dwell Test - Case M - SWA - 115 degrees...... 62 4-17 Sine With Dwell Test - Case M - Control Allocation Effectiveness...... 62
A-1 Validation with Steering wheel angle amplitude of 100 degrees...... 70 A-2 Validation with Steering wheel angle amplitude of 110 degrees...... 70 A-3 Validation with Steering wheel angle amplitude of 130 degrees...... 71 A-4 Dugoff Tire Model Validation - SWA - 130 degrees...... 71
B-1 Sine With Dwell - Case A - Vehicle Response...... 73 B-2 Sine With Dwell - Case A - Actuator Response - Brake Torques...... 74 B-3 Sine With Dwell - Case A - SWA - 130 degrees...... 74 B-4 Sine With Dwell - Case A - SWA - 130 degrees - Control Allocation Effectiveness 75 B-5 Sine With Dwell - Case A and B comparison - Vehicle Response...... 75 B-6 Sine With Dwell - Case A and B comparison - Actuator Response - Brake Torques 76 B-7 Sine With Dwell - Case B - SWA - 130 degrees...... 76 B-8 Sine With Dwell - Case B - SWA - 130 degrees - Control Allocation Effectiveness 77 B-9 Sine With Dwell - Case C - Vehicle Response...... 77 B-10 Sine With Dwell - Case C - Actuator Response - Brake Torques...... 78 B-11 Sine With Dwell - Case C - SWA - 130 degrees...... 78 B-12 Sine With Dwell - Case C - SWA - 130 degrees - Control Allocation Effectiveness 79 B-13 Sine With Dwell - Case D - SWA - 130 degrees - Control Allocation Effectiveness 79 B-14 Sine With Dwell - Case D - SWA - 130 degrees...... 80 B-15 Sine With Dwell - Case E - SWA - 130 degrees - Control Allocation Effectiveness 80 B-16 Sine With Dwell - Case E - SWA - 130 degrees - Parameter Convergence.... 81 B-17 Sine With Dwell - Case F - Vehicle Response...... 81 B-18 Sine With Dwell - Case F - Actuator Response...... 82 B-19 Sine With Dwell - Case F - SWA - 130 degrees...... 82 B-20 Sine With Dwell - Case F - SWA - 130 degrees - Control Allocation Effectiveness 83 B-21 Sine With Dwell - Case F and G comparison - Vehicle Response...... 83 B-22 Sine With Dwell - Case F and G comparison - Actuator Response - Brake Torques 84 B-23 Sine With Dwell - Case G - SWA - 130 degrees...... 84 B-24 Sine With Dwell - Case G - SWA - 130 degrees - Control Allocation Effectiveness 85
Karan Chatrath Master of Science Thesis List of Figures ix
B-25 Sine With Dwell - Case H - Vehicle Response...... 85 B-26 Sine With Dwell - Case H - Actuator Response...... 86 B-27 Sine With Dwell - Case H - SWA - 130 degrees...... 86 B-28 Sine With Dwell - Case H - SWA - 130 degrees - Control Allocation Effectiveness 87 B-29 Sine With Dwell - Case I - SWA - 130 degrees - Control Allocation Effectiveness 87 B-30 Sine With Dwell - Case I - SWA - 130 degrees...... 88 B-31 Sine With Dwell - Case J - Vehicle Response...... 88 B-32 Sine With Dwell - Case J - Actuator Response - Brake Torques...... 89 B-33 Sine With Dwell - Case J - SWA - 130 degrees...... 89 B-34 Sine With Dwell - Case J - SWA - 130 degrees - Control Allocation Effectiveness 90 B-35 Sine With Dwell - Case J - Brake Actuator Parameter Estimation...... 90 B-36 Sine With Dwell - Case J - Pfeffer Steering Actuator Parameter Estimation... 91 B-37 Sine With Dwell - Case K - SWA - 130 degrees...... 91 B-38 Sine With Dwell - Case K - SWA - 130 degrees - Control Allocation Effectiveness 92 B-39 Sine With Dwell - Case K - Brake Actuator Parameter Estimation...... 92 B-40 Sine With Dwell - Case K - Pfeffer Steering Actuator Parameter Estimation... 93 B-41 Sine With Dwell - Case L - Vehicle Response...... 93 B-42 Sine With Dwell - Case L - Actuator Response - Brake Torques...... 94 B-43 Sine With Dwell - Case L - SWA - 130 degrees...... 94 B-44 Sine With Dwell - Case L - SWA - 130 degrees - Control Allocation Effectiveness 95 B-45 Sine With Dwell - Case M - SWA - 130 degrees - Parameter Convergence.... 95
Master of Science Thesis Karan Chatrath x List of Figures
Karan Chatrath Master of Science Thesis List of Tables
3-1 Simple Example - WLS - CA Tuning Parameters...... 32 3-2 Simple Example - MPCA Tuning Parameters...... 34
4-1 High Level PD Controller as in equation 4-6 - Tuning parameters...... 44 4-2 Tuning Parameters - 4 Actuators - WLS - CA...... 48 4-3 Tuning Parameters - 4 Actuators - MPCA - Simple brake actuator...... 49 4-4 APE Initialization - 4 Actuators - APE + MPCA - Simple brake actuator.... 50 4-5 Tuning Parameters - 5 Actuators - WLS - CA...... 51 4-6 Tuning Parameters - 5 Actuators - MPCA - simple brake and steering actuators. 51 4-7 Tuning Parameters - 5 Actuators - APE+MPCA - Simple Brake Model + Nonlinear Steering Dynamics...... 54
A-1 Vehicle model parameters - Non exhaustive list - From IPG CarMaker...... 69
Master of Science Thesis Karan Chatrath xii List of Tables
Karan Chatrath Master of Science Thesis Acknowledgements
Working on this master thesis has been an eventful experience, to say the least. Looking back, I see that I have grown into a better version of myself, as a student of science and as a person too. In this entire year-long journey, a few people played an important role in my life and a few words of thanks is an inadequate expression of gratitude. My parents and my brother, Varun, have always been a constant source of support and encouragement. I thank them for all of that, their patience, and for their continued faith in me. They are the most important people to me.
I consider myself fortunate to have been supervised by Dr. Barys Shyrokau. With his guid- ance, not only have I become a better student, my interest and proficiency in the area of vehicle dynamics control has grown significantly. His feedback on my work proved to be crucial at every stage. I would also like to express my appreciation for Yanggu Zheng. His insights and critical feedback proved to be helpful. Bright colleagues becoming good friends is a desirable eventuality. I would like to thank Nishant, Vishrut, Damian, Pasquale, Arvind and Anoosh for being great companions during this two-year Master program. When there were times I found myself grappling with a concept, or stuck altogether, one of them always lent a patient ear and offered their inputs.
Studying and work can often turn into a monotonous activity. Some friends that I made in the past year, broke this monotony and gave me something more to look forward to. For this, I would like to thank Alex, Atif, Akshaya, Mehrnoush, Patrick, Maria, Mehdi, Abishek and Hendrig. I would especially like to thank Elina for her kindness and companionship during a very critical phase of my work and my life. My friends Dhruv and Alexandra, apart from being great people to spend time with, offered their inputs while I worked on this report. Each of these individuals are wonderful and it has been a privilege to get to know them. Finally, I would like to thank Dr. Tamas Keviczky and Dr. Bilge Atasoy, for taking time out from their schedule to evaluate my master thesis.
Delft, University of Technology Karan Chatrath July 12, 2019
Master of Science Thesis Karan Chatrath xiv Acknowledgements
Karan Chatrath Master of Science Thesis Chapter 1
Introduction
Advancement is a continuous process. The automotive industry is no exception to this fact. Right from the days of the first use of the pneumatic tire, to recent times where the scien- tific community is discussing the possibility of level 4 or level 5 autonomy [1], the quality of road transportation has come a long way forward. Despite making significant strides towards betterment, some key challenges continue to persist, namely, improvement of safety, comfort, performance and the impact of vehicles on the environment. The scope for improvement in each of these domains is never-ending.
Elaborating on the point of improvement of safety, there are two forms of safety elements that a vehicle can be equipped with. One form is passive safety elements, which comprises of seat belts and Supplementary Restraint Systems (SRS) or airbags. Passive safety elements play the role of mitigating the damage or injury caused to a user in the event of an acci- dent. The other form consists of active safety systems. Active safety elements differ in the sense that vehicles are equipped with such systems with the intent of avoiding accidents alto- gether. It is the development of such systems where knowledge of vehicle dynamics along with control engineering becomes critical. Combining the understanding of vehicle dynamics and control theory has led to the development of driver assistance systems, or Advanced Driver Assistance Systems (ADAS). Examples of ADAS includes functions like Antilock Braking System (ABS), Electronic Stability Control (ESC) and Traction Control System (TCS). The inclusion of ADAS in road vehicles has resulted in a decrease in road accidents and fatalities. However, occurrences of road accidents are still reported. This shows the need for betterment in the state of the art of ADAS.
With the inclusion of such features in a vehicle in addition to its regular equipment, a road vehicle now comprises of several actuators that can control the motion of the vehicle. An- other reason for equipping a vehicle with several actuators is the introduction of redundancy. Redundancy in a system can enhance its overall reliability.
Master of Science Thesis Karan Chatrath 2 Introduction
The primary objective of this thesis is to address the problem of designing a motion control system of a vehicle that comprises of many actuators available for control. More specifically, the task of control system design for an ’over actuated mechanical system’ will be looked into. The notion of an over actuated mechanical system will be formally defined in a subsequent section of this chapter. Furthermore, it is to be noted that the automotive application of focus in this work is the ESC system.
1-1 Introduction to Electronic Stability Control
Electronic stability control (ESC) is also known as Vehicle Dynamics Control (VDC), Electronic Stability Program (ESP) and Vehicle Stability Control (VSC). The theory of such control systems has been developed in detail in [2]. The main purpose of such systems is to prevent the vehicle from spinning or drifting out of control especially when it operates near its limits of handling. The way an ESC system prevents vehicle stability loss is that in the event where the system detects that the vehicle might spin out of control, the system ’steers’ the vehicle back to its desired path. This corrective action generated by the controller is done using the braking system (differential braking) or using steer by wire or by using a combination of both.
Figure 1-1 illustrates how the ESC system works using differential braking. The variable controlled by ESC is the yaw rate of the vehicle. The controller, depending on the situation, computes a desired yaw rate that a vehicle should acquire. This desired value is then com- pared with the actual yaw rate and a corrective yaw moment is calculated. This corrective yaw moment is generated by the controller by actuating the brakes, or steering system. This generated brake force differs on each side of the vehicle. It is this difference in brake force on each side that generates the desired corrective yaw moment about the Center Of Grav- ity (COG) of the vehicle.
Figure 1-1: Working Of ESC Using Differential Braking (Simplified top view of a vehicle)
In other words, if a driver steers the vehicle too aggressively, the ESC system assists the driver
Karan Chatrath Master of Science Thesis 1-2 Over-actuated Mechanical Systems 3 during the dynamic task to ensure that the stability of the vehicle is not lost. This system has proven to be useful in events where the driver is compelled to take sudden turns (to avoid obstacles), which push the vehicle to operate near its limits of handling. The risk of loss of traction in this regime of operation is high and sophisticated control strategies are necessary to ensure that the system behaves as desired.
The ESC system generates a corrective yaw moment by applying brakes or by using steer by wire. This means, to control only one state, i.e. yaw rate, at least four actuators are available for control. This is essentially the concept of an over actuated mechanical system.
1-2 Over-actuated Mechanical Systems
This section gives a formal introduction to the notion of over actuated mechanical systems. Consider a mechanical system of which n states need to be controlled. This mechanical system is equipped with m actuators available to control the n states. Then, the mechanical system is said to be over actuated iff m > n (1-1) The behaviour of mechanical systems is captured using mathematical models. These models are usually a system of nonlinear coupled differential equations of the general form
x˙ = f(x, u) (1-2)
Where, x ∈ IRn and u ∈ IRm. An over actuated system will always satisfy the inequality 1-1. Such systems are frequently encountered when one considers aerospace and marine applica- tions. These applications require several redundant actuators for improved system reliability and to ensure fail-safe operation. As road vehicle development is heading towards an era of automated driving, the dependence on drivers to execute dynamic driving tasks is expected to reduce further. The reliability of vehicle subsystems at such times is of great importance.
Effective use of all actuators using good automatic control strategies is essential. One way of approaching the control system design of such systems is by designing individual controllers for each degree of freedom (DOF) of motion. This approach is less than ideal for a few rea- sons. As a vehicle would have more controllers, the complexity of the system and the overall amount of hardware and instrumentation required to be done on the vehicle increases signifi- cantly. This can also increase the energy consumed by the system. Additionally, the DOF of motion of a vehicle are usually coupled in a nonlinear fashion. Therefore, the control action for one specific state can generate unwanted motions in a different state. This can result in overall performance deterioration of the system. An example of such behaviour can be seen in ESC systems. Differential braking is typically used to stabilise the vehicle. Since the brakes are applied, there is an inevitable and unwanted decrease in the longitudinal velocity of the vehicle.
The states of motion of an over actuated system are coupled. Also, an actuator can influence more than one state of the system. This leads one to the question: Does there exist an ideal
Master of Science Thesis Karan Chatrath 4 Introduction combination of actuator commands that enables a vehicle to move in the desired manner? Addressing this question has been a challenge. An additional challenge gets posed when one considers the limits and dynamics of the actuators. These factors must be considered for effective control system design. A strategy known as Control Allocation (CA) has been de- veloped in literature to address these challenges. The stability control systems developed in this thesis utilises the concept ofCA.
1-3 Control Allocation
Control allocation is a technique to control over actuated mechanical systems. The main function of a control allocator is the coordination of actuators so that the system responds in a manner as desired. Using a control allocation algorithm, the control system design can be separated into subtasks. A high-level controller computes a set of forces and moments re- quired to make a vehicle move as wanted. These high-level commands (also known as virtual control inputs) are then fed into the control allocator which in turn distributes them among the available actuators in an optimal way. This working is illustrated in figure 1-2.
Figure 1-2: Control allocation general strategy
The control allocation technique has some interesting features. Firstly, since the control sys- tem is separated into two tasks, the high-level control system is independent of any actuator configurations. In other words, the nature of the actuators has no effect on the high-level control algorithm. This introduces modularity in the control system. TheCA algorithm also ensures that all actuators are utilised in the best way possible. Finding the ’best possible’ set of actuator commands is usually done so by solving an optimization problem online. There exist a plethora of methods by whichCA can be carried out. A survey carried out in [3] provides an insight into many of these methods.
Additionally, the limits of actuation are also considered in theCA problem. It also enables a designer to account for secondary constraints such as energy and fuel consumption. Among many of the methods addressed in literature, many involve formulating theCA problem as a convex optimization problem. The advantage of doing so is that convex subroutines guar- antee global optimal solutions which converge quicker as compared to other optimization strategies [4].
In a subsequent chapter, theCA problem will be formulated and some of the key methods will be described. An assessment of the advantages and shortcomings of the methods will be described which will be followed by a description of how the drawbacks can be overcome.
Karan Chatrath Master of Science Thesis 1-4 Problem Definition 5
Most of the mathematical details will not be presented in this chapter, however, one key equation is highlighted. Recall the formal definition of over actuated systems in relations 1-1 and 1-2. To solve theCA problem, it is essential to have the knowledge of the relation between the virtual control input vector (v) and the actuator commands (u) (refer figure 1-2). The virtual control inputs are high-level commands generated for each state to be controlled. Typically, a linear mapping of the following form is derived.
v = Bu (1-3)
Here v ∈ IRn and u ∈ IRm and of course m > n. The matrix B is known as the control effec- tiveness matrix. It can be seen that B ∈ IRn×m. The details and insights into this equation will be elaborated later. Equation 1-3 is an under determined set of equations which may have a unique solution, may have infinite solutions or may have no solution at all. The CA algorithm is designed such that it either finds this unique solution, or the ’best’ among the infinite solutions or a vector u such that Bu is as close as possible to v, in some sense, in the event where there is no solution to the set.
1-4 Problem Definition
The technique of control allocation and its applicability to vehicle dynamics control is in- vestigated. Based on this, the problem is defined as a goal which is to achieve effective coordination of the actuators of different configurations in order to ensure that the vehicle behaves in the desired manner. This thesis also aims to address the problem of considering actuator dynamics and its uncertainties in the control allocation problem.
1-5 Summary Of Work Done
In the context of vehicle stability control, the vehicle is the over actuated system where the application of control allocation becomes warranted. It was mentioned earlier that manyCA methods involve solving an optimization problem. These optimization problems are usually convex in nature and enable accounting for the limits of actuator operation. This is an im- portant feature ofCA.
However, the behaviour of actuators, just like many electro-mechanical systems are dynamic in nature. In other words, if a command is sent to an actuator, it responds to that com- mand, but not instantaneously. The actuator takes a finite amount to time to build up the appropriate response. In more technical terms, each actuator has some transient dynamics and/or some internal delays. Many of theCA methods investigated in literature operate under the assumption that the actuators respond ’quickly’ to a command. As a result of this assumption, severalCA approaches neglect actuator dynamics altogether.
It will be demonstrated that neglecting actuator dynamics is a risky assumption. A method to overcome this shortcoming has been discussed in detail and it is based on the strategy
Master of Science Thesis Karan Chatrath 6 Introduction of Model Predictive Control (MPC). [5]. The MPC based control allocation technique will be explained in detail in a later chapter. For now, the understanding that MPC is an op- timization based control strategy that operates with a knowledge of the system dynamics is sufficient. The MPC based control allocation technique, or, Model Predictive Control Allo- cation (MPCA), effectively tackles actuator dynamics. However, it relies on the assumption that the dynamics of the actuators are known. However, a situation may so arise where this is not the case. This can be encountered frequently in practice where the behaviour of the actuators is not fully understood and captured accurately by a set of governing differential equations. This creates uncertainties in the actuator dynamics. It is therefore imperative to address this problem. With this understanding, the work done is summarised in the following paragraphs.
Using the concepts of vehicle dynamics and control allocation theory, ESC systems are de- signed. Stability controllers are designed for two actuator configurations (Differential braking and Differential braking + active front steering (AFS)). Among many of the methods that do not account for actuator dynamics, the Weighted Least Squares (WLS)CA method [6] is chosen to carry out preliminary investigations. The reason for doing so is because the WLS -CA method is simple to implement and computationally efficient, as will be described in a later chapter. The performance of the system is assessed with and without the presence of actuator dynamics. The presence of actuator dynamics results in an expected deterioration of closed loop vehicle performance. This establishes the need to resort to methods that account for actuator dynamics.
The MPCA strategy considers actuator dynamics. MPCA based ESC systems are designed (for each actuator configuration) in order to investigate the features of the technique and the performance of the vehicle. MPCA operates under the assumption that actuator dynamics are accurate. The situation where actuator dynamic uncertainties exist is also investigated. The shortcoming of handling actuator uncertainties is addressed by combining the MPCA algorithm with an online parameter estimator. This parameter estimator is adaptive in na- ture and fits the input-output behaviour of actuators to a linear system of known order and mathematical structure. The estimated parameters of this linear system are in turn used by the MPCA technique to solve the control allocation problem.
The dynamics of the actuators used in this thesis are of a simplified nature. In other words, the actuator dynamics considered are governed by linear differential equations. This is a simplification. The next step is to test the parameter estimation technique combined with MPCA, on a set of actuators of an unknown nonlinear nature. In other words, the actuator dynamics are no longer governed by transfer functions but are of a more complex nature. Based on the results of the simulations, conclusions are drawn and an appropriate course of action for the future is recommended.
Karan Chatrath Master of Science Thesis 1-6 Contributions 7
1-6 Contributions
The main contribution of this work is the application of model predictive control allocation for vehicle stability control. An extension to theCA design is carried out by combining the MPCA solver with an online adaptive parameter estimator to address uncertainties in actuator dynamics.
1-7 Layout of This Master Thesis
This Master thesis has been divided into five chapters. Chapter 1 gives a brief introduction to the concepts of over actuated mechanical systems and control allocation. It also introduces the idea behind vehicle dynamics control and its importance for passenger and driver safety. The chapter also articulates the goal of this master thesis and summarises the work carried out.
Chapter 2 goes into the details of vehicle dynamics modelling. In order to develop any con- trol system, an understanding of the system to be controlled is necessary. Since the task of control allocation is to ensure effective coordination of actuators to fulfill a control demand, the chapter also focuses on the description of the actuators. Chapter 3 provides the theoret- ical foundation for control allocation. A brief literature review is carried out followed by a detailed description of the CA methods used in this thesis. The features and shortcomings of each of these methods is highlighted by presenting a simple illustrative example for the readers.
Chapter 4 finally addresses the application of control allocation for electronic stability control for road vehicles. All methods described in chapter 3 are applied to develop ESC controllers for vehicles. The investigations demonstrate the development of controllers for two actuator configurations: Brakes only and Brakes with Active front steering. This chapter addresses many scenarios, the results of most of which are summarised in an appendix. In Chapter 5, this thesis is finally concluded by highlighting some of the key results and by drawing necessary conclusions. The scope for taking this work forward is presented to the readers.
Master of Science Thesis Karan Chatrath 8 Introduction
Karan Chatrath Master of Science Thesis Chapter 2
Vehicle Dynamics Modelling
Simulations are an integral part of automotive research and development activities. In order to carry out meaningful simulations, it becomes necessary to use models of vehicles that accurately depict how their dynamic behaviour is in reality. The models that are used for investigations can be of various degrees of complexity. There exist vehicle dynamics models ranging from one to two degrees of freedom (DOF) to complex multi-body models that can be well over a hundred DOF. Work has been carried out over a span of several decades as a result of which, the models in use by the industry today exhibit a high degree of complexity and accuracy. For the purpose of this thesis, three vehicle dynamics models have been used. They are:
• Multi-body vehicle model in IPG CarMaker. • 7 Degree of freedom planar vehicle dynamics Model. • 2 Degree of Freedom linear bicycle model
The subsequent sections of this chapter describe each of these models. The main objective of this work is motion control. In order to perform motion control, a plant to be controlled is required. The multi-body vehicle model serves as the plant model. The controllers are based on the simplified planar and bicycle models due to their relatively simple and sufficient mathematical description.
2-1 IPG CarMaker Multi Body Vehicle Model
The IPG CarMaker software is a vehicle simulation software. It contains a complete model of the environment comprising of a driver model, a detailed multi-body vehicle model, and models for roads, traffic, and other driving conditions. This provides a user with a suitable platform to build and simulate a wide range of test scenarios [7].
Master of Science Thesis Karan Chatrath 10 Vehicle Dynamics Modelling
Elaborate details of the vehicle model and vehicle subsystem models can be found in the software documentation [8]. The software can be used along with other simulation tools such as MATLAB and Simulink. The software offers a lot of flexibility as it enables users to in- corporate their own models for vehicle subsystems into the simulation environment.
While working with the multi-body vehicle model, it is assumed that all vehicle, tire and subsystem signals can be measured and these measurements are noise-free. This assumption eliminates the need to design state estimators. All investigations are carried out on a flat road and external disturbances like side winds and road unevenness are considered to be absent.
2-2 Planar Vehicle Model
The planar vehicle model is a 7 degree of freedom vehicle model which captures longitudi- nal, lateral, and yaw motions of the car. This section aims to provide a detailed description of the planar vehicle dynamics model. More information about this model can be found in [9].
2-2-1 Vehicle Body
Figure 2-1: Planar vehicle motion indicating coordinate frames
The planar model also incorporates the simplified rotational dynamics of each wheel. Degrees of freedom such as roll, pitch, and vertical motion are neglected. The effects of unsprung
Karan Chatrath Master of Science Thesis 2-2 Planar Vehicle Model 11 masses are also neglected. The vehicle is modelled as a single rigid body with tires. Vehicle planar motion is analysed from two frames of reference. The inertial frame of reference is denoted as G in figure 2-1 while the body-fixed frame has its origin at the COG of the vehicle and is denoted as B. Vehicle parameters (even those not mentioned in figure 2-1) include the following:
• Tf and Tr are the front and rear track widths respectively.
• Lf and Lr are the longitudinal distances between the COG and front and rear axles respectively.
• hcg is the height of the center of gravity from the ground.
• δ is the steering angle of the front wheels.
• ψ is the heading or yaw angle. Its derivative ψ˙ is the yaw rate of the vehicle body.
• β is the body slip angle which will be defined later in this section.
• vx and vy are the longitudinal and lateral components of velocity respectively, defined in the body fixed frame.
• Fxi and Fyi are the longitudinal and lateral forces generated by the tire in its own frame of reference. Fzi are the individual wheel loads. Here, i ∈ {fl,fr,rl,rr}.
• m is the mass of the vehicle and the moment of inertia about its local Z-axis is Izz.
The motion of the vehicle although observed from an inertial frame of reference is analysed with respect to the body-fixed frame of reference. The velocity components of the vehicle expressed in the inertial frame are found by inspecting figure 2-1. They are
" # " #" # V cos(ψ) − sin(ψ) v gx = x (2-1) Vgy sin(ψ) cos(ψ) vy
The acceleration components in the global frame can be obtained by computing the time derivatives of Vgx and Vgy. These acceleration components are Agx and Agy respectively. Having carried out this step, the accelerations of the vehicle body expressed in the body-fixed frame can be computed as such:
" # " #" # a cos(ψ) sin(ψ) A x = gx (2-2) ay − sin(ψ) cos(ψ) Agy
Master of Science Thesis Karan Chatrath 12 Vehicle Dynamics Modelling
The symbols ax and ay are the longitudinal and lateral acceleration components respectively. By working out and simplifying equations 2-1 and 2-2, one obtains
ax = v˙x − ψv˙ y ay = v˙y + ψv˙ x (2-3) The equations of motion of the vehicle body are formulated using Newton-Euler Equations. m v˙x − ψv˙ y = Fx m v˙y + ψv˙ x = Fy
Izz ψ¨ = Mz (2-4) The right-hand sides of equation set 2-4 can be derived by analysing the free body diagram of motion of the vehicle in the body-fixed frame of reference. Looking at figure 2-1, the expressions for the net forces and moments are: Longitudinal motion:
Fx = F xrl + Fxrr + Fxfl cos (δ) + Fxfr cos (δ) − Fyfl sin (δ) − Fyfr sin (δ) (2-5)
Lateral motion:
Fy = Fyrl + Fyrr + Fyfl cos (δ) + Fyfr cos (δ) + Fxfl sin (δ) + Fxfr sin (δ) (2-6)
Yaw motion:
F T F T M = xrr r − xrl r − (F + F )L + F L cos (δ) + F L cos (δ)− z 2 2 yrr yrl r yfl f yfr f F T cos (δ) F T cos (δ) xfl f + xfr f + F L sin (δ) + F L sin (δ)+ (2-7) 2 2 xfl f xfr f F T sin (δ) F T sin (δ) yfl f − yfr f + M 2 2 za
Mza is the total self-aligning moment contributed by each tire. For the purpose of the work conducted in this thesis, this term is neglected in further investigations.
Another quantity is of interest in the study of vehicle dynamics is that of the body slip angle. It is the angle between the direction the vehicle is heading, to that of the resultant vehicle velocity. It quantifies the ’slip’ or lateral motion that a vehicle experiences. It is denoted by the Greek letter β and is defined as v β = tan−1 y (2-8) vx
Karan Chatrath Master of Science Thesis 2-2 Planar Vehicle Model 13
2-2-2 Wheel and Tire Related Quantities
Single Corner Wheel Dynamics
This section describes the rotational dynamics of each wheel. The wheel dynamics are cap- tured by using the single corner model [10] which is described as follows
Jwω˙ wi = Tdi − Tbi − Fxirw (2-9)
Here, i ∈ {fl,fr,rl,rr}. Tdi is the drive torque and Tbi is the brake torque at the i − th wheel. The effective radius of the wheel is rw. Jw is the rotational inertia of the wheel.
Wheel Loads
Due to the vehicle accelerating in both longitudinal and lateral directions, the normal load Fzi acting at the point of contact of each wheel varies with time. The wheel loads are given by the following equations.
L F = mg r − F − F zfl 2L z,long z,lat,f L F = mg r − F + F zfr 2L z,long z,lat,f L F = mg f + F − F zrl 2L z,long z,lat,r L F = mg f + F + F zrr 2L z,long z,lat,r (2-10)
Here, g is the acceleration due to gravity, L = Lf + Lr, hcg is the height of the center of gravity of the vehicle from the ground, and
ma h F = x cg z,long 2L mayLrhcg Fz,lat,f = Tf L mayLf hcg Fz,lat,r = TrL (2-11)
Here, the effects of unsprung masses and the static roll stiffness are ignored.
Slip Angles
The pneumatic tire is not a rigid body. As the wheel turns, the tire carcass deforms. Due to this deformation the direction in which the wheel heads and its velocity vector are offset by
Master of Science Thesis Karan Chatrath 14 Vehicle Dynamics Modelling an angle. This angle is known as the slip angle. This has been discussed in [9]. Here, only the equations are presented. ˙ −1 vy + Lf ψ αfl = δ − tan ˙ Tf ψ vx − 2 ˙ −1 vy + Lf ψ αfr = δ − tan ˙ Tf ψ vx + 2 ˙ −1 vy − Lrψ αrl = − tan Trψ˙ vx − 2 ˙ −1 vy − Lrψ αrr = − tan Trψ˙ vx + 2 (2-12)
Longitudinal Slip Ratio
Another useful concept used extensively in tire modelling is that of the longitudinal slip ratio. It is defined as a normalized relative wheel velocity defined as for two cases, assuming small tire slip angles. It is denoted by κ. For accelerating wheel: ω r − v κ = w w xw (2-13) ωwrw
For braking wheel:
v − ω r κ = xw w w (2-14) vxw
2-3 Tire Modelling
Tires play a critical role in vehicle motion. These elements are directly in contact with the road and govern a lot of dynamic characteristics of a road vehicle. It is, therefore, necessary to accurately model tire behaviour. These days, several very sophisticated models have been developed that take into account many transient and nonlinear effects. IPG CarMaker itself contains detailed descriptions of various tires. However, in this thesis, highly simplified tire models have been used in subsequent sections (for the process of control system design only). One is the linear tire model while another is the physics-based Dugoff model.
2-3-1 Linear Tire Model And Friction Circle
The linear tire model is the simplest tire model. When one considers pure lateral and pure longitudinal slip, the longitudinal tire force is a function of the longitudinal slip ratio while
Karan Chatrath Master of Science Thesis 2-4 The Linear Bicycle Model 15 the lateral force is a function of the slip angle. At low slip ratios and low slip angles, the tire forces developed can be assumed to vary linearly as such
Fxi = Cκiκi (2-15)
Fyi = Cαiαi (2-16)
Here, Cα is known as the cornering stiffness of the tire while Cκ is referred to as the longi- tudinal slip stiffness of the tire. Here, i ∈ {fl,fr,rl,rr}. The concept of the friction ellipse or friction circle is also important as it quantifies the limits of tire force generation. The simplified friction circle is captured using the relation:
2 2 2 Fxi + Fyi ≤ (µFzi) (2-17)
2-3-2 Dugoff Tire Model
This is a physical model derived in [11]. This model takes into account longitudinal and lateral tire behaviour. It does not compute other quantities like self-aligning moments. It also considers the limits of tire force generation by using the concept of the friction ellipse. The equations governing this model are
Cκiκi Fxi = f (λ) (2-18) 1 − κi
Cαi tan αi Fyi = f (λ) (2-19) 1 − κi Where,
(λ(2 − λ) λ < 1 f(λ) = (2-20) 1 λ ≥ 1 µ F (1 − κ ) λ = n zi i (2-21) q 2 2 2 (Cκiκi) + (Cαitan(αi)) q 2 2 µn = µ 1 − ervxi κi + (tan (αi)) (2-22)
2-4 The Linear Bicycle Model
The planar vehicle model described earlier can be further simplified by making some assump- tions. They are: The tire model is linear and the vehicle is considered to be moving at a constant longitudinal velocity. This vehicle model is used extensively for simplified vehicle motion analysis and control system design.
Master of Science Thesis Karan Chatrath 16 Vehicle Dynamics Modelling
A linear dynamical 2nd order system can be derived, of the form
x˙ = Ax + Bδδ h iT x = vy ψ˙
−(Cαf +Cαr) −L C +L C f αf r αr − v mvx mvx x A = 2 2 −Lf Cαf +LrCαr −(Lf Cαf +LrCαr) Izvx Izvx T h Cαf Lf Cαf i Bδ = m Iz (2-23)
Note that: Cαf = Cαfl + Cαfr is the front axle cornering stiffness of the vehicle and Cαr = Cαrl +Cαrr is the rear axle cornering stiffness of the vehicle. All necessary vehicle parameters are defined in appendixA.
2-5 Actuator Dynamics
No actuator has an instant response to a command. They have a transient response and maybe some internal delays which need to be modelled for the purpose of vehicle dynamics control. Not accounting for the transient response of the actuators hinders a control system to perform to its full potential. In this section, the brake and front steering actuator dynamics will be briefly described.
2-5-1 Brake Actuator Dynamics
Simplified Brake Model
The brake system in consideration during this work is actuated hydraulically. A typical hydraulic brake system consists of a master cylinder attached to the brake pedal which is controlled by a driver. Additional brake pressure can be applied by a controller. The pressure developed in the master cylinder is transmitted to the brake calipers at each wheel. This relation between the input pressure to the brake system and the actual caliper pressure is captured using a simple first-order transfer function model. This model is of the form presented in [12]
P (s) 1 actual = e−δts (2-24) Pcommanded (s) τs + 1 The brake pressure can be converted into a brake torque using a simple static relationship of the form Tbrake = PP 2M Pbrake (2-25)
In many of the simulations reported later, this simplified brake model is used. Here, τ = 1/20, δt = 0.01s. The parameter PP 2M = 11.25. The maximum pressure that can be developed in
Karan Chatrath Master of Science Thesis 2-5 Actuator Dynamics 17 the brake system is 160 bar. The rate of brake torque build up is 12000Nm/s and the rate of brake torque release is 8000Nm/s. These numbers can be converted into rates of pressure build up and release by using the factor PP 2M .
Hydraulic Brake Model
Figure 2-2: Hydraulic brake model schematic for a single wheel [8]
For testing the control system in a more realistic environment, a hydraulically actuated brake model (HAB) has been used. The model presented in this section captures a larger number of details as opposed to the simplified model.
The complete hydraulic model consists of two brake circuits. Each circuit actuates a pair of wheels. There can be two configurations of the circuits. One is the ’X’ configuration where a circuit operates a front-wheel brake and a rear-wheel brake on the other side. In the other configuration, each circuit applies the brakes for each axle (front and rear). A schematic of the primary circuit for one wheel brake is shown in figure 2-2.
In figure 2-2, there are two sets of valves, namely, the inlet and outlet valves. The inlet valve regulates the build-up of pressure in the brake system while the outlet valve regulates the
Master of Science Thesis Karan Chatrath 18 Vehicle Dynamics Modelling release of pressure. Both valves operate independently of each other and are not the same. The opening and closing of each of these valves is what is regulated by the ESC system logic.
2-5-2 Steering Actuator Dynamics
Simplified Steering Model
Typically, a static relation between steering wheel input and wheel steering angle exists. This relation is defined by the so called steering ratio which is
δswa Sratio = (2-26) δwheel The steering system behaviour is captured using simplified second order dynamics: This dynamic system is of the form:
2 δactual (s) ωn −δts = 2 2 e (2-27) δcommanded (s) s + 2ζωns + ωn In this simplified description of the steering system, the control action is aimed at assisting the driver as opposed to automating the steering. This is known as active front steer (AFS). The limits of the active front steer actuation are −30◦ to 30◦, and the rates of the actuator ◦ ◦ are limited between 50 /sec and 50 /sec. Additionally, ωn = 30 rad/sec and ζ = 0.7 and δt = 0.007s.
Pfeffer Steering Model
In comparison to the 2nd order dynamic steering model, the Pfeffer model is more detailed. The system comprises of two units namely the mechanical module and the power assistance module. The mechanical module comprises of the steering wheel, the torsion bar, the rack and pinion, and other mechanical linkages. The model takes into account friction and damping effects. The assistance module is what enables power steering functionality. The assistance can be either offered hydraulically or electronically. While using this model during investiga- tions, the Electronic power steering (EPS) assistance module is enabled. The model is within IPG CarMaker and more elaborate details of it can be found in [8]. A schematic of the Pfeffer model can be studied in figure 2-3.
Karan Chatrath Master of Science Thesis 2-6 Dynamic Driving Manoeuvre 19
Figure 2-3: Mechanical Module of Pfeffer Steering Model with All assistance modules except EPS [8]
2-6 Dynamic Driving Manoeuvre
The dynamic manoeuvre chosen to evaluate the control systems in this thesis is the standard ISO Sine With Dwell (SWD) test [13–15]. This is suitable to test vehicle stability when pushed near the limits of handling. The Sine With Dwell steering input can be seen in figure 2-4. It consists of three-quarters of a 0.7Hz sine wave with a 0.5-second dwell. The dwell starts 1.07 seconds after the beginning of steering. The asymmetry of the steering profile promotes the vehicle to spin at larger steering angles. During this test, the initial longitudinal speed of the vehicle is maintained at 80km/hr.
The performance metrics include a few criteria such as the lateral displacement at 1.07 sec- onds after the beginning of steer must be greater than 1.83m. Additionally, the yaw rates after 1s and 1.75 seconds after completion of steering must not exceed 20% and 35% of the peak yaw rate respectively.
Master of Science Thesis Karan Chatrath 20 Vehicle Dynamics Modelling
Figure 2-4: Sine With Dwell Test Steering Input
2-7 Validation Of The Planar Model
The planar vehicle model, as mentioned earlier, is used for the purpose of control system design. To validate the planar model, the standard ISO Sine with Dwell Test is chosen. This test is used mainly to evaluate stability control systems. The validation is done by assuming that the tire forces and front-wheel steering angle are available measurements (Obtained from IPG CarMaker). Using these virtual sensor measurements, the model behaviour is assessed. The vehicle is given an initial speed of 80km/hr. Four variants of the steering wheel angle amplitude are tested namely 100◦, 110◦, 120◦ and 130◦. The complete validation of the vehicle and Dugoff tire model can be found in AppendixA. Here, only the case for 120◦ is presented. Refer to figures 2-5 and 2-6.
In order to use the Dugoff tire model, knowledge of quantities such as slip angle and longi- tudinal slip ratio are necessary. It is assumed that the virtual sensors in IPG CarMaker that generate these signals are accurate and noise-free. This assumption eliminates the need for state estimation. The other validation results are reported in appendixA.
Karan Chatrath Master of Science Thesis 2-7 Validation Of The Planar Model 21
Figure 2-5: Validation with Steering wheel angle amplitude of 120 degrees
Figure 2-6: Dugoff Tire Model Validation for lateral forces - Steering Wheel Angle 120 degrees (TM - Dugoff Tire Model)
Master of Science Thesis Karan Chatrath 22 Vehicle Dynamics Modelling
2-8 Summary
This chapter presented details of the modelling of the dynamics of a vehicle and its subsystems. Specifically, three models were described. One is a multi-body vehicle model available in IPG CarMaker. The second model described in detail is the seven DOF planar vehicle model. Then, a simple linear bicycle model was presented to the readers. The simplified planar and bicycle models are used during control system design and the IPG multi-body model is considered as a plant model. This thesis is on control allocation, which is a strategy involving the coordination between actuators. A description of the steering and brake actuators used for investigations has also been provided. This chapter concludes with a validation of the planar vehicle model.
Karan Chatrath Master of Science Thesis Chapter 3
Control Allocation Theory
3-1 Control Allocation Problem Formulation
A brief introduction to control allocation (CA) was provided in section 1-3 of this report. This chapter aims to describe this strategy in more detail. Control allocation is a technique used for over actuated mechanical systems. The use of a control allocator divides the control system design into subtasks:
• A high-level controller computes a set of forces and moments to control the motion. This controller is designed independent of actuator configurations.
• This control action generated by the high-level controller, or control demand is dis- tributed among all available actuators using the control allocator, typically by solving an optimization problem.
Figure 1.2 demonstrates this strategy. The mathematical description is as follows. Consider a nonlinear dynamic system described by the following general state-space description.
x˙ = f (x, u) x ∈ IRn u ∈ IRm (3-1)
Here, u is a vector of all actuator commands, while x is a vector of states to be controlled. Recall, that the inequality m > n is the necessary condition, if satisfied, qualifies a system as over actuated. Without any loss of generality, the dynamic system can be re-written as follows:
x˙ = f (x) + g (x) v (3-2) And v = h (x, u) (3-3)
Master of Science Thesis Karan Chatrath 24 Control Allocation Theory
Here, v is a vector known as a virtual control input vector and can be mapped to the vector of actuator commands, where v ∈ IRn. This mapping is of a nonlinear nature and using equation 3-3 to solve the control allocation problem would amount to solving a nonlinear programming (NLP) problem. Real-time computational efficiency of such NLP algorithms is a problem that is avoided by linearising the equation 3-3 about a suitable operating point ue. Using a first-order Taylor expansion:
! ∂h v ≈ v (h, ue) + (u − ue) (3-4) ∂u u=ue
Typically, no actuator commands produce no control inputs. For this reason, ue is taken as the zero vector. This leads to the relationship
v = Bu (3-5)
Where, B ∈ IRn×m.
∂h B = (3-6) ∂u u=ue This matrix is known as the control effectiveness matrix. It was mentioned in chapter 1 that the control system design becomes modular with the introduction of CA. This can be seen in equation 3-2 that the state space description is independent of all actuator commands. The task of the high-level controller is to compute the vector v. Looking back at equation 3-5, we see an underdetermined set of equations which may have a unique solution, many have infinite solutions or may have no solution. The task of the control allocation algorithm is to find this unique solution, or the best among the infinite solutions or a solution for the vector u such that Bu is as close to v, in the event where there are no solutions.
The above brief formulation of the control allocation problem has been discussed in detail in [6, 16]. A feature of CA is that it accounts for the limits of the actuators. This ensures that the solution obtained does not compel the actuators to operate outside their range of operation.
3-2 Control Allocation Methods - A Brief Literature Review
There are several methods using which the control allocation problem can be solved. Methods have been devised and worked on since the early ’90s. A lot of them were developed keeping in mind aerospace and marine applications. In those systems, each subsystem has many levels of actuator redundancy to ensure overall reliability. Earlier primitive methods involved ensuring that the actuator commands generated by the control allocator do not violate their constraints. Examples of such methods are: • Explicit Ganging • Redistributed pseudo-inverse • Daisy chaining A detailed description of these methods can be found in [17] with simple examples to illustrate each method. The downside of such methods is that they only ensure that the actuators do
Karan Chatrath Master of Science Thesis 3-2 Control Allocation Methods - A Brief Literature Review 25 not violate their constraints. The solutions obtained in these cases are quite often not the ’best’, or in other words, ’optimal’.
In the work carried out by Durham and Bordignon [18–22], the authors provide a geometri- cal interpretation of the control allocation problem. Looking at equation 3-5 and keeping in mind the actuator constraints, they called the set of all feasible solutions of the CA problem, u, as the set of ’admissible control actions’. The corresponding set of virtual control input vectors was referred to as the ’attainable moment set’. The authors devised a method to ensure that the CA problem yielded solutions that fall within the set of admissible control actions. This approach, in the year 2002, was reformulated as a linear programming program by Bodson [16]. This method came to be known as the method of direct control allocation.
Apart from the mentioned methods, a lot of work was carried out to formulate and solve the CA problem as a constrained optimization problem. Notable contributions were made by Härkegård in his Ph.D. thesis [6]. In his work, the author mainly focused on formulating the control allocation problem as a convex optimization problem. More specifically, the focus of that work was to formulate the CA problem as a constrained least squares problem.
Solving optimization problems online can be computationally expensive. The work done in [23] lays emphasis on the algorithms used to solve 2-norm based optimization problems. The method of solving the optimization problem was a computationally and numerically ef- ficient implementation of the active set method [24, 25]. In addition to developing efficient algorithms to solve 2-norm based CA problems, the author of [6] also made a MATLAB based toolbox known as ’Quadratic Programming Control Allocation Toolbox’ [26]. Other similar toolboxes for solving CA problems have been described in [27, 28].
The algorithms available for use in the openly available QCAT toolbox are: • Sequential Least Squares (SLS) • Weighted Least Squares (WLS) • Minimal Least Squares (MLS) • Interior point method based WLSCA • Cascaded generalised inverses (CGI) • Fixed point algorithm • Direct control allocation • Dynamic control allocation There is also an algorithm combining optimal control and control allocation. Details of this work can be found in [29]. The importance of convex optimization techniques, that are nu- merically and computationally efficient, is known. Slow convergence to solutions can give rise to time delays in a closed-loop control system, resulting in possible deterioration of overall system performance.
In addition to developing efficient CA techniques, its also becomes important to consider the health of the actuators themselves. This thought led towards development in the domain of fault tolerant control allocation. These techniques enable the CA algorithm to deal with
Master of Science Thesis Karan Chatrath 26 Control Allocation Theory unforeseen actuator deterioration or failures. To briefly summarise such techniques, the CA algorithm is constantly ’informed’ about the health of the actuators, and using this knowl- edge, makes necessary changes if actuator failures or deterioration is detected. Work relevant to fault-tolerant control allocation techniques has been carried out in [30–38].
On some occasions, it becomes necessary to account for the nonlinear relationships between virtual control inputs and actuator commands, as described in equation 3-3. Framing the CA problem as an optimization problem would result in a nonlinear and non-convex optimiza- tion problem. Such CA problems come under the category of nonlinear control allocation techniques. Relevant work in this domain has been carried out in [39–42]. Framing the CA problem as a nonlinear optimization can offer a designer a lot of flexibility. It enables users to account for additional terms in the cost function such as minimization of fuel consump- tion, energy efficiency, structural loads, etc. Such an atypical choice of cost functions can be found in the work carried out in [43, 44]. The CA problem can also be framed as a multi- objective optimization problem [45]. The downside of nonlinear and multi-objective-nonlinear CA methods is that the problem is most likely non-convex.
An important assumption is made in many of the CA techniques highlighted in this brief re- view of the literature. The assumption is that the actuators respond fast to a command, and as a result, actuator dynamics can be neglected. The following section aims to describe, in detail, the Weighted Least Squares CA algorithm, which also operates under this assumption, just like most other methods.
3-3 Weighted Least Squares Control Allocation
It was mentioned earlier that the control allocation algorithm takes into account the limits of the actuators. Let us consider a set of actuators with the following position and rate limits. The constraints are formulated as per the work carried out in [6].
uMin ≤ u(t) ≤ uMax (3-7)
ρMin ≤ u˙(t) ≤ ρMax (3-8) The rate limits can be re written as position limits by approximating the derivative as follows. u (t) − u(t − T ) u˙ (t) ≈ (3-9) T Based on this, the overall actuator constraints are
u ≤ u ≤ u (3-10)
Where
u = max[uMin, u (t − T ) + T ρMin]
u = min[uMax, u (t − T ) + T ρMax] (3-11)
Karan Chatrath Master of Science Thesis 3-4 Dealing With Actuator Dynamics 27
The CA problem is now considered. According to the work carried out in [6], the CA problem can be formulated as a two-step optimization problem that involves the minimisation of two 2-norm cost functions (Sequential Least Squares). This two-step problem can be merged into a single optimization problem of the form
2 2 u = arg min kWu(u − ud)k2 + γkWv(Bu − v)k2 (3-12) u≤u≤u
Here, ud is the desired set of actuator commands (which is typically set to zero), Wv and Wu are positive definite weighting matrices using which the emphasis laid on each DOF or actuator command can be varied as tuning parameters. γ is a parameter, which is typically taken to be a large value since the desired outcome of the control allocator should be such that equation 3-5 is satisfied. This method is the WLS algorithm. It is a single step optimization procedure which makes it numerically faster when compared to the sequential least squares algorithm. This method is also found to be quite frequently used in literature, specifically for vehicle dynamics control [46–49]. The algorithm used to solve the quadratic programming problem defined in equation 3-12 is known as the active set method. It has been programmed to be computationally efficient [23].
The WLS CA method is one among several methods that do not consider actuator dynamics. A typical block diagram representation of the WLS strategy can be studied in figure 3-1. Figure 3-1 can be considered applicable to any related CA method that does not account for actuator dynamics, and not just the WLS CA method.
Figure 3-1: Weighted Least squares CA block diagram
3-4 Dealing With Actuator Dynamics
A general block diagram of a CA algorithm where actuator dynamics have been introduced can be studied in figure 3-2. The reason behind neglecting actuator dynamics in the WLS method is that the dynamics of the actuators are much faster in comparison to the response time of the dynamical system.
Master of Science Thesis Karan Chatrath 28 Control Allocation Theory
Figure 3-2: Control allocation with actuator dynamics considered
The WLS method can indeed operate effectively when actuators respond fast. However, when the dynamics of the actuators are relatively slower, the WLS loses effectiveness. Since many CA methods lack the ability to handle actuator dynamics, alternate approaches that over- come this shortcoming become necessary. The subsequent section goes into details of one such method.
3-5 Model Predictive Control Allocation
MPCA is a control allocation technique based on model predictive control and it effectively accounts for the actuator dynamics. This section of the chapter is dedicated to the theory of MPCA. Model predictive control is an optimal control technique based on the idea of receding horizon control. The theory behind MPC is developed in detail in [5]. The following paragraphs briefly explain how MPC works.
The controller can be formed based on a linear or nonlinear model of the plant. This math- ematical model of the plant is used to look ahead in time and understand how the plant is expected to evolve with time. This mathematical description is also known as the prediction model and the time duration for which it looks ahead is known as the prediction horizon.
After having computed the prediction model and predicted the future throughout the pre- diction horizon, a set of control inputs are computed for a certain number of time steps into the future. This number of time steps is known as the control horizon. This computation is carried out by solving an optimization problem. This optimization problem reflects certain control objectives and is usually convex in nature. MPC allows the designer to account for constraints on the inputs and outputs. After having solved the optimization problem, from the resulting set of control inputs that span the control horizon, only the first element of the control input vector is applied to the plant. This process is repeated for each time step as the plant’s behaviour evolves.
Karan Chatrath Master of Science Thesis 3-5 Model Predictive Control Allocation 29
MPCA works in a similar fashion, except here, the dynamics behaviour captured is that of the actuators. The theory of MPCA has been developed based on [50] and [5]. The rest of this section is dedicated to the formulation of the MPCA problem. Consider the following systems of actuators of an over actuated mechanical system. The number of actuators is considered to be Nu, and each actuator is modelled as a continuous-time, linear dynamical system. The dynamics of the i-th actuator is of the form
x˙ i(t) = Aixi(t) + Biuci(t − δi)
ui = Cixi (3-13)
Here, xi represents the states of the i-th actuator, uci represents the command sent to the i-th actuator, and ui is the response of the i-th actuator. Here, i ∈ (1, 2, 3 ...Nu). Since the MPCA algorithm operates in discrete time, each of these actuators can be discretized according to the zero-order hold (ZOH) operation. The resulting discrete-time dynamics of each actuator, without any loss of generality, can be represented as
xi (k + 1) = Adixi (k) + Bdiuci(k)
ui(k) = Cdixi(k) (3-14)
The actuator dynamics can be combined into a single state-space description as such
x (k + 1) A 0 ... 0 x (k) u (k + 1) 1 d1 1 B ... 0 c1 x (k + 1) 0 A ... 0 x (k) d1 u (k + 1) 2 d2 2 . . . c2 . = . . . + . .. . . . . . .. . . . . . . . . 0 . . 0 ...BdNu xNu (k + 1) 0 0 ...AdNu xNu (k) ucNu (k + 1) u1(k) Cd1 0 ... 0 x1(k) u2(k) 0 Cd2 ... 0 x2(k) . = . . . . . . .. . . . . . 0 . uNu (k) 0 0 ...CdNu xNu (k) (3-15)
Equations 3-15 can be written in short as
x (k + 1) = Adx (k) + Bduc (k)
u (k) = Cndx(k) (3-16)
Equation set 3-16 describes the combined discrete-time actuator dynamics. Recall that v = Bu. Using this, the following is obtained
x (k + 1) = Adx (k) + Bduc (k)
u (k) = Cndx(k)
BCnd = Cd (3-17)
Master of Science Thesis Karan Chatrath 30 Control Allocation Theory
Finally, the combined state-space representation of the actuator dynamics is
x (k + 1) = Adx (k) + Bduc (k)
v(k) = Cdx(k) (3-18)
Ns×Ns Ns×Nu No×Ns The sizes of each system matrix is: Ad ∈ IR ,Bd ∈ IR , Cd ∈ IR . For formulating the MPCA problem, the notation shown above will be used consistently. To begin the formulation of the problem, three key pieces of information are necessary. The sampling time of the control system, the prediction horizon Np and the control horizon Nc. Knowing this, the equation 3-19 can be iterated for future time steps as shown
v (k + 1) = CdAdx (k) + CdBd uc (k) 2 v (k + 2) = CdAdx (k) + CdAdBd uc (k) + CdBd uc(k + 1) . . Np Np−1 Np−Nc v (k + Np) = CdAd + CdAd Bd uc (k) + ··· + CdAd Bd uc (k + Nc − 1) (3-19) This set of iterated equations can be combined in a matrix form as such v(k + 1) CdAd CdBd ... 0No×Nu uc(k) C A2 C A B ... 0 v(k + 2) d d d d d No×Nu uc(k + 1) . = . x (k) + . . . . . . . . . . . ... . . Np Np−1 Np−Nc v(k + Np) CdAd CdAd Bd ...CdAd Bd uc(k + Nc − 1) (3-20) Here, the notation 0p×q represents a matrix of p rows and q columns and having all elements as zero. The equation 3-20 can be rewritten in a condensed form
V (k) = F x (k) + φ Uc(k) (3-21) Equation 3-21 is known as the predictor model based on which the controller operates. The matrix F is of a similar form as the observability matrix and the matrix φ is known as the Toeplitz matrix. The sizes of each of these matrices and vectors are V (k) ∈ IRNpNo×1, NpNo×Ns NpNo×NuNc NuNc×1 F ∈ IR , φ ∈ IR and Uc(k) ∈ IR . The next step in the formulation of the MPCA problem is accounting for constraints imposed on the actuators. Recall that the constraints on the actuators comprise of position as well as rate limits. The rate constraints can be re-written as position constraints as in equation 3-11. The vector variable in the MPCA optimization problem is the vector Uc. The constraints need to be recast in terms of that. All the actuator commands computed through MPCA must satisfy all constraints throughout the control horizon Nc. This is done as such h iT Ain = −INu×Nc INu×Nc h iT bL = uc uc . . . uc h iT bU = uc uc ... uc h iT bin = −bL bU (3-22)
Karan Chatrath Master of Science Thesis 3-6 A Simple Example 31
The relations in equation 3-22 can be combined to yield a combined set of inequality con- straints AinUc ≤ bin (3-23) Having computed the predictor model, as per equation 3-22 and the constraints on the MPCA problem as per equation 3-23, the next step is to define the control allocator objective. Let us consider a high-level controller to be generating a virtual input vector or control demand namely vref . The control allocator must compute a set of actuator commands such that the actuators together produce this control demand throughout the prediction horizon. This can be captured by doing the following. The reference virtual control demand is defined referred to as Vref . h iT Vref = vref vref . . . vref (3-24)
NpNo×1 Here, Vref ∈ IR . Now, the MPCA optimization is formulated as such. It is formed such that it is a convex optimization problem. The cost function is defined as
T T J = (V (k) − Vref ) W (V (k) − Vref ) + Uc (k) QUc(k) (3-25)
The objective is to minimise the cost function J with respect to the vector Uc such that the constraints defined in equation 3-23 are satisfied. Equation 3-21 is replaced in the cost function and expanded leading to the formation of a convex quadratic programming problem. This concludes the formulation of the MPCA problem. The MPCA problem has several tun- ing parameters. There is the sample time of the control allocator T and the prediction and control horizons and the diagonal elements of the weighting matrices W and Q.
3-6 A Simple Example
In this section, a simple example is presented to the readers. This example is aimed at demonstrating certain features and possible shortcomings. of the CA algorithms used in this thesis. A system comprising of two actuators is considered as such:
u˙ 1 = −2u1 + 2u1c
u˙ 2 = −u2 + u2c (3-26)
The position and rate limits of each actuator are taken to be the same in this example. The position is limited between −20 ≤ ui ≤ 20. The rate is limited between −500 ≤ u˙ i ≤ 500. Here i ∈ {1, 2}. The scenario here is that the system is equipped with a high-level controller and a control allocator. The hypothetical high-level controller works such that it produces a time-varying virtual control demand v which is of the form
v = 10 sin(t) (3-27)
The system is over actuated since a single control demand v has to be optimally distributed among two actuators. Here, the control effectiveness matrix is taken as
B = [1 2] (3-28)
Master of Science Thesis Karan Chatrath 32 Control Allocation Theory
At first, the weighted Least squares control allocator is tested on this system without con- sidering actuator dynamics according to figure 3-1. The tuning parameters for the control allocator are as per table 3-1.
Tuning Parameter Value Control Allocation Effectiveness weight - Wv 1 Actuator Command Weight - Wu diag[0.1;0.1] Relative Weight - γ 1 Sampling Time of CA 0.01 sec Max. Number of Iterations 100
Table 3-1: Simple Example - WLS - CA Tuning Parameters
The control allocator, in this case, works effectively as can be seen in figure 3-3. The effec- tiveness of the control allocator is assessed by how well the actuators can together produce the required demand. In other words how close is Bu to v.
Figure 3-3: Simple example - WLS - CA - No actuator Dynamics
Without actuator dynamics considered, the control allocator works effectively as seen in the figure above. On the introduction of actuator dynamics (according to figure 3-2), the effectiveness of the control allocator can be assessed in figure 3-4. In this case, the control allocator’s tuning parameters are kept the same as the previous case where actuator dynamics were not considered. It can be seen that the introduction of actuator dynamics causes a deterioration in the effectiveness of the control allocator. This is because the WLS method neglects actuator dynamics.
Karan Chatrath Master of Science Thesis 3-6 A Simple Example 33
Figure 3-4: Simple example - WLS - CA - With actuator Dynamics
The next step of this simple example is to test the model predictive control allocation algo- rithm. Keeping in mind that the system is equipped with actuators described in equations 3-26, the tuning parameters for the MPCA algorithm can be studied in table 3-2.
Figure 3-5: Simple example - MPCA
In table 3-2, the weight matrices W and Q need to be resized according to the prediction and control horizons. The effectiveness of the MPCA method can be assessed in figure 3- 5. Figure 3-5 demonstrates that MPCA can handle actuator dynamics. However, MPCA operates under one assumption. The assumption is that the actuator dynamics are known and that there are no large uncertainties in the actuator dynamics.
Master of Science Thesis Karan Chatrath 34 Control Allocation Theory
Tuning Parameter Value Control Allocation Effectiveness Weight - W 1 Actuator Commands Weight - Q diag([0.01;0.01]) Prediction Horizon - Np 50 Control Horizon - Nc 20 Sampling Time of CA 0.01 sec
Table 3-2: Simple Example - MPCA Tuning Parameters
3-7 MPCA With Adaptive Parameter Estimation (APE)
It was mentioned that the MPCA algorithm operates under the assumption that the actuator dynamics are known. This section tests a situation where this is not the case. This test was carried out by operating under the premise that the MPCA algorithm is formulated based on the set of actuator dynamics described as
u˙ 1 = −20u1 + 20u1c
u˙ 2 = −70u2 + 70u2c (3-29)
The actual actuator parameters are according to those described in equations 3-26. All tuning parameters are according to table 3-2. In figure 3-6, it is seen that there is a reduction in the effectiveness of the control allocator. The actuators do manage to meet the demand v but this happens after a visible delay. This shortcoming of handling actuator uncertainties will be addressed in this section by combining the MPCA algorithm with a parameter estimator which is adaptive in nature.
Figure 3-6: Simple example - MPCA with actuator uncertainties
Karan Chatrath Master of Science Thesis 3-7 MPCA With Adaptive Parameter Estimation (APE) 35
3-7-1 Online adaptive parameter estimation
The work carried out in the area of system identification and parameter estimation has been of great interest in the past decades and continues to be a topic that researchers embark on. In this thesis, the parameter estimation algorithm used is the Auxiliary Model Based Recursive Least Squares (AM-RLS) Algorithm. Details of this work can be found in [51,52]. The algorithm is described in this section. Consider a general discrete-time transfer function describing some form of input-output relationship of the form m m−1 m−2 Y (z) 1 a0z + a1z + a2z + ··· + am = d n n−1 n−2 (3-30) U(z) z z + b1z + b2z + ··· + bn Here, d is a known delay and the variables m and n are also known. The coefficients of each power of the variable z are considered unknown. This transfer function can be re-written as such: n n−1 n−2 z + b1z + b2z + ··· + bn Y (z) = (3-31) m−d m−1−d m−2−d −d a0z + a1z + a2z + ··· + am U(z) Now, both sides are multiplied by a filter of the form 1/(z + λ)n to give
zn zn−1 zn−2 1 + b + b + ··· + b Y (z) = (z + λ)n 1 (z + λ)n 2 (z + λ)n n (z + λ)n (3-32) zm−d zm−1−d zm−2−d z−d a + a + a + ··· + a U(z) 0 (z + λ)n 1 (z + λ)n 2 (z + λ)n m (z + λ)n Now, taking the inverse Z-transform on both sides yields filtered versions of the signals y(k) and u(k). The filtered versions of the signals obey the following notation
zn−iY (z) Z−1 = y (k) (z + λ)n fi zm−d−iU(z) Z−1 = u (k) (z + λ)n fi (3-33)
Here, λ is a number which is chosen by the user and its magnitude is taken to be within the unit circle centered at the origin. With this in mind equation 3-31 is converted into time domain giving the following
yf0(k) + b1yf1(k) + b2yf2(k) + ··· + bnyfn(k) = a0uf0(k) + a1uf1(k) + a2uf2(k)+ (3-34) ··· + amufm(k) After a bit of rearrangement
z(k) = φT (k)θ(k) h iT φ(k) = −yf1(k) ... −yfn(k) uf0(k)(k) . . . ufm(k) h iT θ(k) = b1 b2 . . . bn a0 a1 . . . am (3-35)
Master of Science Thesis Karan Chatrath 36 Control Allocation Theory
The above equation 3-35 is referred to as a linear-in-the-parameters parametric model. Here, z(k) is the signal yf0(k), which, like φ(k), is also computed at each instant of time. It is based on this parametric model that parameter estimation is carried out. The estimate of the parameter vector at the present time instant is referred to as θˆ(k). Having explained the parametric model, the AM-RLS algorithm is presented as follows [51, 52] θˆ(k) = θˆ(k − 1) + P (k)φ(k) z(k) − φT (k)θˆ(k − 1)
P (k − 1)φ(k)φT (k)P (k − 1) P (k) = P (k − 1) + 1 + φT (k)P (k − 1)φ(k) (3-36)
The initial conditions of the algorithm can be defined by setting P (0) to be a larger number like 100 or 1000, and the initial parameter estimate θˆ(0) can be set to smaller values (0.1, 0.01, etc.). There are a number of parameter estimation methods that exist in literature that make use of the parametric model in 3-35. Typically, such estimators are derived by minimizing some kind of a cost function. What distinguishes different algorithms is the structure of the cost function and the resulting speed of parameter convergence of the algorithm. The AM-RLS algorithm, just like many others operates under a few assumptions. They are:
• Only the input and output of the system to be estimated are measured and all mea- surements are noise-free.
• The order of the system, number of zeros and the input delay of the system are consid- ered to be known quantities.
• The input to the system is persistently exciting. It is this condition which is necessary and sufficient to ensure asymptotic convergence of parameters.
This adaptive parameter estimation algorithm is henceforth abbreviated as ’APE’ for the rest of this report. This parameter estimator is combined with the MPCA algorithm. This combination is henceforth referred to as ’APE+MPCA’ or ’APE with MPCA’.
3-7-2 Combining APE with MPCA
The APE algorithm fits input-output data to transfer functions with a known structure and delay. In the event where the actuator dynamics to be considered in MPCA are uncer- tain, it can be combined with the APE to overcome this uncertainty. This combination of APE+MPCA is tested on the simple example described earlier. In this case, all tuning pa- rameters of the MPCA are defined according to table 3-2. Additionally, the initial values of the matrix P and the vector θˆ are set to 100I2 where I2 is the identity matrix of size 2, and h iT 0.1 0.1 respectively. The effectiveness of this control allocation algorithm can be assessed in figure 3-7. It is seen that the shortcoming of handling actuator uncertainties is effectively handled by the MPCA algorithm combined with the parameter estimator.
The parameter convergence can be studied in figure 3-8. The reader is asked to observe the values to which the parameters converge. If one were to take the dynamics defined in
Karan Chatrath Master of Science Thesis 3-7 MPCA With Adaptive Parameter Estimation (APE) 37 equation 3-26, and perform a zero-order hold discretization with a sampling time of 0.01s, the parameters of the resulting first-order discrete transfer functions would be the same as the ones seen in figure 3-8.
Figure 3-7: Simple example - APE with MPCA
Figure 3-8: Simple example - APE with MPCA - Parameter Convergence
Master of Science Thesis Karan Chatrath 38 Control Allocation Theory
3-8 Summary
This chapter laid out the underlying theory of control allocation. The problem was formu- lated and a brief review of literature was carried out. Among all methods briefly discussed, the weighted least squares algorithm was elaborated in detail and its shortcomings were high- lighted. Keeping the shortcomings in mind, the theory of model predictive control allocation was developed. Both these methods were tested on a simple example based on which each of their features and shortcomings was demonstrated. After establishing that MPCA is not as effective when actuator dynamic uncertainties are present, the MPCA algorithm is combined with an adaptive parameter estimator.
These three approaches of control allocation, namely, weighted least squares, MPCA and APE+MPCA is what this entire thesis is based on. Using these allocation strategies, vehi- cle yaw rate stability control systems are designed and evaluated on simulation. A detailed analysis of these control systems is the subject of the subsequent chapters of this report.
Karan Chatrath Master of Science Thesis Chapter 4
Electronic Stability Control Using Control Allocation
This chapter delves into the domain of vehicle dynamics control. All the theory built up in the previous chapters leads to this application. The vehicle is stabilised by controlling its yaw rate, using differential braking or active steering or a combination of both. A basic introduction to vehicle stability control is given in the introductory chapter of this thesis. This chapter, on the other hand, goes into the details and results of the work carried out. Vehicle dynamics control has been covered in several publications. A lot of the theory applied in this thesis is inspired by literature. Noteworthy references are that of Laine [46–49] and Shyrokau [53,54].
While assessing the controllers designed in this chapter on simulation, the vehicle is subjected to the standard ISO Sine With Dwell test (refer section 2-6). Before going into the details of the control system design, a need for vehicle yaw rate control is established in the following section.
4-1 Sine With Dwell Test With No Control
In this section, the performance of the vehicle, with no stability controller, when subjected to the Sine With Dwell (SWD) Test is studied, using IPG CarMaker. The SWD test is per- formed by varying the steering wheel angle amplitudes (SWA) for a given initial vehicle speed of 80 km/hr. The vehicle performance can be studied in figure 4-1.
It can be seen that the vehicle loses stability beyond 110 degrees of SWA. There is a consid- erable loss of vehicle speed and the yaw rate does not stabilize to zero after completion of the steering input. It can also be seen that the vehicle experiences high lateral accelerations of nearly 0.8 g’s. Such levels of accelerations indicate that the vehicle is operating near its limits
Master of Science Thesis Karan Chatrath 40 Electronic Stability Control Using Control Allocation of handling. An aggressive steering input of 115 degrees SWA causes the vehicle to spin out of control.
Figure 4-1: Vehicle Response With No Control
The necessity to control the stability of the vehicle is therefore established. Loss of lateral control also manifests itself in the form of high body slip angles. This quantity indicates that the vehicle acquires high lateral velocity which needs to be regulated appropriately. The work conducted in this thesis is, however, confined to yaw rate control.
4-2 General Layout Of The Stability Control System
This section aims to give a brief layout of the control system. A general block diagram of the closed-loop system can be studied in figure 4-2. It can be seen that the plant model is the vehicle model within IPG CarMaker. The goal is to control the Yaw rate of the plant model. The overall control system has been divided into three subsystems which are: the reference generator, the high-level controller and the control allocation algorithm.
The goal of the reference generator is to generate a signal that represents the desired yaw rate that the vehicle must acquire. The high-level controller performs the computation of a corrective yaw moment in the event where it detects a difference between actual and desired yaw rates of the vehicle. The control allocation algorithm has the task of distributing the high level control action among available actuators in such a way that the response of the actuators when combined amounts to the corrective yaw moment demanded by the high level controller.
Karan Chatrath Master of Science Thesis 4-3 Reference Generator 41
Figure 4-2: General Block Diagram for ESC Using Control Allocation
Subsequent sections of this report will elaborate on each of these subsystems in detail.
4-3 Reference Generator
The yaw rate to be controlled requires a desired signal to which the controller drives it. The reference yaw rate can be obtained in more than one way. In this report, two approaches are reported. One approach is shown as follows and is based on the linear bicycle model.
x˙ = Ax + Bδδ h iT x = vy ψ˙ (4-1)
The linear bicycle model is a highly simplified 2-DOF vehicle model described in equation 4-1. The system matrices are shown in chapter 2, equation 2-23. The reference signal can be generated based on this model. However, since the model is simplified, the peak reference yaw rate computed by the linear bicycle model does not account for the limitations of the road conditions. The coefficient of friction must be factored into this calculation and the upper and lower bound of the yaw rate is
˙ µg ψbound = (4-2) vx
Here, µ is the friction coefficient and vx is the longitudinal velocity of the vehicle. This sequence of computations based on the linear bicycle model does limit the yaw rate. However, the resulting reference signal has some ’abrupt changes’ or ’sharp corners’. This can have an adverse impact on passenger comfort as the control system tends to follow this ’desired’ signal and abrupt changes can lead to higher jerk. To avoid this, another approach is used which generates a smooth curve as the reference yaw rate signal. This second approach also involves the use of the bicycle model, however, in this case, only its steady-state response is considered.
Master of Science Thesis Karan Chatrath 42 Electronic Stability Control Using Control Allocation
Looking at equation 4-1, one can compute the steady-state yaw rates and lateral velocities by equating its derivatives to zero as such
T h i −1 xss = vyss ψ˙ss = −A Bδδ (4-3)
The steady-state yaw rate expression is then limited according to equation 4-2. This is then filtered through a standard second-order transfer function of appropriate frequency and damping. This filter is designed so that the filtered steady state yaw rate response is as close as possible to that of the linear bicycle model. The reference trajectories predicted by both approaches can be seen in figure 4-3. These reference signals in figure 4-3 are generated for the sine with dwell test. The signals will be different for different manoeuvres are they depend on the driver steering input.
Figure 4-3: Reference signals for yaw rate control for the sine with dwell test
The natural frequency and damping ratio of this second-order filter is ωn = 15 rad/sec and ζ = 0.7. It can seen in figure 4-3 that the steady-state filtered response is much smoother than that predicted by the linear bicycle model.
Karan Chatrath Master of Science Thesis 4-4 High-Level Controller 43
4-4 High-Level Controller
It was explained in previous chapters that by using control allocation, the task of control system design can be separated into sub-tasks. One is the high-level control while the other is the control allocation algorithm. The high-level controller is independent of any details of the actuators and is therefore independent of the control allocation algorithm. Consider the equations of motion of a planar vehicle as described in equation 2-4. This is rewritten here as such
m 0 0 v˙x mψv˙ y Fx ˙ 0 m 0 v˙y = −mψvx + Fy (4-4) 0 0 Izz ψ¨ 0 Mz
This can be wriiten as
Mx˙ = f(x) + v (4-5)
Equation 4-5 is clearly devoid of any actuator commands. The task of the high-level controller is to compute the vector v of corrective forces and moments. In this thesis, different control allocation algorithms are used. However, the high-level controller parameters in all these cases are kept constant (irrespective of the number of actuators used and control allocation algorithm applied) in order to compare the results objectively. Note that
h iT v = Fx Fy Mz (4-6)
Since the control objective is that of yaw rate, the high-level controller computes a corrective yaw moment for the vehicle. In other words
h iT v = 0 0 Mz,corr (4-7)
The high-level controller is kept as simple as necessary. The term f(x) in equation 4-5 is neglected during control for this reason. The choice of the controller is a standard discrete- time proportional-derivative regulator operating at 100 Hz. The proportional controller is meant to drive the yaw rate response towards the reference signal while the derivative action is used to ensure that the response is adequately damped. Tuning of this controller was done by trial and error and its form is:
N C(z) = Kp + K (4-8) d NTs 1 + z−1
The error signal is defined as e(k) and its Z transform is denoted by E(z)
˙ ˙ e(k) = ψref (k) − ψ(k) (4-9)
Master of Science Thesis Karan Chatrath 44 Electronic Stability Control Using Control Allocation
˙ ˙ Here ψref (k) and ψ(k) are discrete-time yaw rate signals sampled at 100 Hz. k is the present time instant. The corrective yaw moment is computed as follows:
−1 Mz,corr(k) = Z [C(z)E(z)] (4-10)
h iT The high-level control action is therefore v(k) = 0 0 Mz,corr(k) . The controller tuning parameters are kept constant for all simulations reported for vehicle stability control analysis. They are summarised in table 4-1. These parameters were obtained after tuning the closed- loop control system depicted in figure 4-2.
Tuning Parameter Value Proportional Control - Kp 9000 Derivative Control - Kd 1000 Filter Coefficient - N 1
Table 4-1: High Level PD Controller as in equation 4-6 - Tuning parameters
The controller has been tuned so that the vehicle can be stabilized up to a steering wheel am- plitude of 130 degrees. This is a limited range of operation, however, the objective remains to evaluate the control allocation strategies as opposed to controlling the vehicle for more aggressive steering inputs. Even in the range where the SWA is between 110 degrees and 115 degrees, the vehicle experiences high lateral accelerations and a considerable amount of body slip (figure 4-1). Controlling the vehicle, albeit in this limited range, is of great importance.
4-5 Control Effectiveness Matrix Derivation
Recall that in previous chapters, it was mentioned that in order to use a control allocation algorithm, a mapping between states to be controlled and actuator commands is necessary. Such a mapping is often linear and of the form described in equation v = Bu. B is the control effectiveness matrix. This section derives the mapping between planar motion and available actuators. Consider the planar vehicle model described in section 2-2. If one were to neglect the effect of steering angles in the equations of motion (described in equations 2-5, 2-6, 2-7) by approximating sin (δ) = 0,cos (δ) = 1, they simplify to.
Fx = Fxrl + Fxrr + Fxfl + Fxfr
Fy = Fyrl + Fyrr + Fyfl + Fyfr T T M = (F + F ) L − (F + F ) L + f (−F + F ) + r (−F + F ) z yfl yfr f yrl yrr r 2 xfl xfr 2 xrl xrr (4-11)
Consider the single corner model as described in equation 2-9. If one were to neglect the wheel angular accelerations, the model gets reduced to a steady-state relationship which is
Tdi − Tbi Fxi = (4-12) rw
Karan Chatrath Master of Science Thesis 4-5 Control Effectiveness Matrix Derivation 45
Equation 4-12 can be replaced in equation set 4-11. Additionally, the linear tire model (section 2-3-1) can be used to replace the lateral forces. The small-angle approximation is applied to the slip angle equations. The resulting equation is what is finally used to establish a relationship between the forces/moments and actuator commands. Equation set 4-11 can be written generally as v = h (x, u) (4-13) Where x is a state vector and u is the actuator command vector. The following cases are now considered.
4-5-1 Case 1: 4 Actuators - Differential Braking
The number of high level control actions is three and the number of available actuators is h iT h iT four. In other words, v = Fx Fy Mz and u = Tbfl Tbfr Tbrl Tbrr .
Using 4-13, one can obtain the control effectiveness for this case by carrying out the following operation
∂h B = (4-14) ∂u This gives the mapping v = Bu, which is
T F − 1 − 1 − 1 − 1 bfl x rw rw rw rw T F = 0 0 0 0 bfr (4-15) y T Tf Tf Tr Tr brl Mz − − 2rw 2rw 2rw 2rw Tbrr
4-5-2 Case 2: 5 Actuators - Differential Braking And Active Front Steering
The number of high level control actions is three and the number of available actuators is five. h iT h iT In other words, v = Fx Fy Mz and u = Tbfl Tbfr Tbrl Tbrr δ . By carrying out the same steps as done in the previous case, one can obtain the required mapping v = Bu, where is control effectiveness matrix is:
− 1 − 1 − 1 − 1 0 rw rw rw rw B = 0 0 0 0 Cαf (4-16) T T f − f Tr − Tr C L 2rw 2rw 2rw 2rw αf f
Here, Cαf = Cαfl + Cαfr is the front axle cornering stiffness of the vehicle. This mapping developed (in each case) between states and actuator commands is essential for the control allocation as will be seen in subsequent sections.
Master of Science Thesis Karan Chatrath 46 Electronic Stability Control Using Control Allocation
4-6 Summary Of All Simulation Scenarios
The signals generated by the high-level controller, or the virtual input, or the control demand, are fed into the control allocation algorithm. The task of the control allocation algorithm is the coordination of available actuators such that the control demand is met. In the process of carrying out yaw rate control by employing control allocation methods, several scenarios with different algorithms have been evaluated on simulation. This section aims to clearly present all the scenarios considered. The simulations have been carried out by increasing the complexity of the control allocation algorithm in a progressive manner. As mentioned earlier, two actuator configurations have been considered. One configuration uses four actuators (Differential braking) while the other uses braking and active front steering.
4-6-1 Configuration 1: With Four Actuators
The control effectiveness matrix for this actuator configuration is presented in equation 4-15. The following scenarios are assessed.
• Case A: Weighted Least squares with no actuator dynamics
• Case B: Weighted least squares algorithm with simplified brake actuator dynamics
• Case C: MPCA with simplified brake actuator dynamics
• Case D: MPCA with simplified brake actuator dynamics and with actuator uncertainties
• Case E: APE+MPCA with simplified brake actuator dynamics
4-6-2 Configuration 2: With Five Actuators
The control effectiveness matrix for this actuator configuration is presented in equation 4-16. The following scenarios are assessed.
• Case F: Weighted Least squares with no actuator dynamics
• Case G: Weighted least squares algorithm with simplified brake and steering actuator dynamics
• Case H: MPCA with simplified brake and steering actuator dynamics
• Case I: MPCA with simplified brake and steering actuator dynamics and with actuator parameter uncertainties
Karan Chatrath Master of Science Thesis 4-7 Details Of Simulation Scenarios and Results 47
4-6-3 Additional Scenarios
It can be seen that from cases A to I, only simplified actuator dynamics are used for investi- gations. The following scenarios incorporate either the Hybdrulic brake model as described in section 2-5-1 or the Pfeffer Steering model as described section 2-5-2. Additionally, some practical constraints and considerations from a vehicle dynamics standpoint are considered for some cases. The practical considerations include the following:
• Incorporation of the limits of tire force generation in the control allocation constraints.
• Modifying the control effectiveness matrix so as to make it dynamic in nature.
The additional cases considered are:
• Case J: APE+MPCA using five actuators. Here the brake dynamics are simple as per section 2-5-1. However, the steering model in use is the relatively complex Pfeffer Model (see section 2-5-2).
• Case K: APE+MPCA using five actuators. Here the brake dynamics are simple as per section 2-5-1. However, the steering model in use is the Pfeffer Model. The limits on tire force generation and the dynamic control effectiveness matrix are also incorporated in this analysis.
• Case L: WLS using four actuators. In this scenario, the hydraulic brake model is used instead of the simplified brake actuator dynamics. The limits on tire force generation and the dynamic control effectiveness matrix are also incorporated in this analysis.
• Case M: APE+MPCA using four actuators. In this scenario, the hydraulic brake model is used instead of the simplified brake actuator dynamics. The limits on tire force generation and the dynamic control effectiveness matrix are also incorporated in this analysis.
The scenario of using five actuators, all of which are not described by simplified dynamics was attempted, however, the simulations crashed. In all, there are 13 scenarios which have been named as Cases A to M. Each of these scenarios are discussed in subsequent sections. Most of the simulation results are presented in appendixB. It should be noted that the high level PD controller has the same set of tuning parameters in all cases listed above.
4-7 Details Of Simulation Scenarios and Results
The underlying theory behind control allocation has been discussed in chapter 3 in detail. Specifically, the weighted least squares algorithm and the model predictive control allocation algorithm were described. Combining the MPCA algorithm with an adaptive parameter esti- mation technique was also elaborated upon. These techniques are all used for vehicle stability control.
Master of Science Thesis Karan Chatrath 48 Electronic Stability Control Using Control Allocation
Recall that in chapter 2, two actuator configurations were defined. One configuration com- prises of four actuators (brakes only) while the other is a combination of brakes with active front steer. The control effectiveness matrices for each of these cases have been derived and are found in equations 4-15 and 4-16. For most of the simulations carried out, the simplified dynamics of the brake and steering actuators were used as described in 2-5. The limits of the brake actuators are defined as follows
uMax = 1800 Nm
uMin = 0 Nm
ρMax = 12000 Nm/sec
ρMin = −8000 Nm/sec
uMin ≤ uc(t) ≤ uMax
ρMin ≤ u˙c(t) ≤ ρMax (4-17)
The steering system dynamics is described using a second-order transfer function described in section 2-5-2. The limits of the active front steering steering actuation are
◦ uMax,AF S = 30 ◦ uMin,AF S = −30 Nm ◦ ρMax,AF S = 50 /sec ◦ ρMin,AF S = −50 /sec
uMin,AF S ≤ uc(t) ≤ uMax,AF S
ρMin,AF S ≤ u˙c(t) ≤ ρMax,AF S (4-18)
4-7-1 Tuning Parameters
The tuning parameters for the WLS-CA algorithm can be studied in table 4-2. The general block diagram for WLSCA can be seen in figure 3-1.
Tuning Parameter Value Control Allocation Effectiveness weight - Wv diag[0;0;1] Actuator Command Weight - Wu diag[0.1;0.1;0.1;0.1] Relative Weight - γ 1 Sampling Time of CA 0.01 sec Max. Number of Iterations 100
Table 4-2: Tuning Parameters - 4 Actuators - WLS - CA
The reason for choosing Wv to be larger than Wu is because a greater emphasis needs to be placed on reference tracking than limiting the magnitude of actuator commands. γ is typically chosen to be very high values like 106. However, here, a nominal value is chosen as a larger value may result in unnecessarily high brake torques, in the process of minimising
Karan Chatrath Master of Science Thesis 4-7 Details Of Simulation Scenarios and Results 49 the error between v and Bu. The sample time of the control allocator is kept similar to that of the high-level controller. A lower sampling time would result in many more instances where the optimization problem needs to be solved. This would lead to an increase in online computational load. The tuning parameters for the MPCA algorithm can be seen in table 4-3. Tuning Parameter Value Control Allocation Effectiveness Weight - W diag[0;0;1] Actuator Commands Weight - Q diag([0.1;0.1;0.1;0.1]) Prediction Horizon - Np 30 Control Horizon - Nc 10 Sampling Time of CA 0.01 sec
Table 4-3: Tuning Parameters - 4 Actuators - MPCA - Simple brake actuator
Figure 4-4: MPCA BLOCK Diagram - General
The weight matrices W and Q need to be resized according to the prediction and control horizons. Lower sampling time of a model predictive control allocator enables it to effectively reject unknown disturbances [55]. On the other hand, a low sampling time for the CA also increases the computational load. In the simulations reported, the selected scenarios are de- void of unknown disturbances, so the sampling time is taken to be the same as that of the high-level controller.
A rule of thumb for the choice of prediction horizon is that it should be chosen so that it can look ahead throughout the transient response of the slowest actuator [55]. This was con- sidered while tuning the controllers. The control horizon should ideally be kept as low as possible. This is because the number of variables for the optimization problem reduces, thus promoting faster computations and convergence. However, small values like Nc = 2 do not enable the system to respond in the desired way. The control horizon was increased up to a point where the response of the vehicle and the control allocator is considered acceptable. A general block diagram for the MPCA algorithm can be seen in figure 4-4.
Master of Science Thesis Karan Chatrath 50 Electronic Stability Control Using Control Allocation
The third control allocation technique involves the combination of APE and MPCA. The tun- ing parameters of the MPCA solver are kept the same as table 4-3. The parameter estimator, in this case, works on estimating two unknown parameters of a first-order transfer function with a known time delay. The APE algorithm requires initialization of two quantities as per equation 3-36. One initialization is that of the initial estimate of the unknown parameter vector θˆ(0) while the other is the matrix P (0). The initial conditions for the APE can be seen in the following table.
Tuning Parameter Value θˆ(0) [0.1;0.1] P(0) 100I2
Table 4-4: APE Initialization - 4 Actuators - APE + MPCA - Simple brake actuator
Figure 4-5: APE+MPCA BLOCK Diagram - General
It should be noted that in table 4-4, the symbol I2 denotes the identity matrix of size two. A general block diagram where APE is combined with MPCA can be studied in figure 4-5. It has been described earlier that the APE requires knowledge of the order of the system, the delay, and the input and output of the dynamics to be estimated. The estimated parameters are then fed back into the MPCA algorithm following which the control allocation problem is solved with the knowledge of the estimated actuator dynamics.
The tuning parameters defined in tables 4-2, 4-3, 4-4 are mainly defined for the four actuator configuration. The following tables show the tuning parameters when five actuators are in use. The tuning parameters for the WLS algorithm can be seen in table 4-5.
Karan Chatrath Master of Science Thesis 4-7 Details Of Simulation Scenarios and Results 51
Tuning Parameter Value Control Allocation Effectiveness weight - Wv diag[0;0;1] Actuator Command Weight - Wu diag[0.1;0.1;0.1;0.1;5000] Relative Weight - γ 1 Sampling Time of CA 0.01 sec Max. Number of Iterations 100
Table 4-5: Tuning Parameters - 5 Actuators - WLS - CA
The tuning parameters for MPCA can be studied in table 4-6. The prediction and control horizons are chosen according to the description given earlier in this section. A very large weight is placed on the AFS action. The reason behind doing so is the same as that done while using the WLS algorithm.
Tuning Parameter Value Control Allocation Effectiveness weight - W diag[0;0;1] Actuator Command Weight - Q diag([0.1;0.1;0.1;0.1;108]) Prediction Horizon - Np 30 Control Horizon - Nc 25 Sampling Time of CA 10−2 sec
Table 4-6: Tuning Parameters - 5 Actuators - MPCA - simple brake and steering actuators
The block diagrams for the MPCA and APE+MPCA strategies can be seen in figures 4-4 and 4-5 respectively.
4-7-2 Simulation Scenario Case E
Most of the results obtained for each scenario as described in section 4-6 are reported in appendixB. In this section, case E is discussed. Recall that case E is the scenario where four simplified brake actuators are considered. The control allocation algorithm used is the APEMPCA technique.
Figure 4-6, shows the performance of the vehicle in the sine with dwell test for a given steering wheel angle amplitude (SWA). It is seen that the vehicle is stabilized for SWA up to 130 degrees. Figure 4-7 shows the corresponding brake torques generated by the ESC controller. Figure 4-8 gives an idea of the performance metrics of the sine with dwell test, as described in section 2-6. The idea here is that if the yaw rate or the lateral displacement curves intersect with the red dashed lines, then the manoeuvre fails. It can be seen in figure 4-8 that the vehicle behaves in a stable manner for a steering input of 130 degrees amplitude.
Master of Science Thesis Karan Chatrath 52 Electronic Stability Control Using Control Allocation
Figure 4-6: Sine With Dwell - Case E - Vehicle Response
Figure 4-7: Sine With Dwell - Case E - Actuator Response
Karan Chatrath Master of Science Thesis 4-7 Details Of Simulation Scenarios and Results 53
Figure 4-8: Sine With Dwell - Case E - SWA - 130 degrees
The control allocation algorithm works effectively in this scenario. The parameter estimation convergence and the control allocation effectiveness can be studied in appendixB.
4-7-3 Simulation Scenario - Case J
This section aims to address cases J as defined in section 4-6. The results in the cases A to I only consider simplified actuator dynamics. The simplified actuator dynamics are linear dynamical systems. It has been seen that the MPCA algorithm when combined with the APE works effectively. Estimating the parameters of a linear system of a known order and structure is a manageable task. However, in reality, things are different. The mathematical structure of the dynamics of the actuators is usually unknown. In fact, the governing dynam- ics of the actuators could be highly nonlinear. In this case, an attempt has been made to ’fit’ the response of an unknown actuator to a linear dynamical system.
Looking at the scenario defined in case J, the goal of this investigation is to see whether the response of the steering system can be fit into a linear dynamical system. More specifically, can the steering response be fit into a first order transfer function? The simulations that follow make use of the tuning parameters described in table 4-7. Recall that the high-level controller is kept constant throughout all cases.
The input to the steering system is the driver input (steering wheel angle) + AFS and the output is the road wheel angle δ. The parameter estimator uses this input-output information to fit the response to a first order transfer function.
Master of Science Thesis Karan Chatrath 54 Electronic Stability Control Using Control Allocation
Tuning Parameter Value Control Allocation Effectiveness Weight - W diag[0;0;1] Actuator Command Weight - Q diag([0.1;0.1;0.1;0.1;106]) Prediction Horizon - Np 30 Control Horizon - Nc 25 Sampling Time of CA 10−2 sec APE parameter: P(0) 10I2 APE parameter: θˆ(0) [0.1 ; 0.1]
Table 4-7: Tuning Parameters - 5 Actuators - APE+MPCA - Simple Brake Model + Nonlinear Steering Dynamics
The results of the simulations carried out for the case J is summarized in AppendixB. How- ever, an important observation is highlighted. When the active front steer is combined with differential braking, the total control demand is distributed among a larger number of actu- ators. As a result, the magnitudes of brake torques generated by the controller are lower. A consequence of this is that the vehicle experiences a lower loss of speed as opposed to when only four actuators are used. One can argue based on this that integrated yaw rate control using brakes and active front steering is a better option for vehicle dynamics control.
Another observation to be noted is that despite the response of the steering system being approximated by a first-order transfer function, the control allocation algorithm still works effectively. The subsequent sections delve into the incorporation of practical considerations from a vehicle dynamics standpoint following which cases K and M are discussed.
4-8 Additional Vehicle Dynamics Factors
In the previous section, a part of the simulation scenarios (as described in section 4-6) is discussed. It was reported that APE combined with MPCA effectively handles actuator dy- namics as well as actuator uncertainties. However, those evaluations carried out were still theoretical in nature. Practical aspects from a vehicle dynamics standpoint need to be taken into consideration.
One such practical consideration is the pneumatic tires. Tires, in general, are limited in their performance. No tire can produce an unlimited amount of force. So while the control alloca- tion algorithm does consider the limits of the actuators, it also needs to take into consideration the limits of the tire. To elaborate a bit further, the brake system aids in the generation of a braking torque in the wheels. This, in turn, results in the generation of longitudinal tire forces that cause the vehicle to decelerate. During an aggressive manoeuvre, a situation may arise where the tire cannot produce the force required to decelerate the vehicle. This could result in a loss of traction with the road, which can be dangerous. Therefore, it is imperative to factor in the limits of the tire in the constraints of the control allocation problem.
The second practical consideration is the variation of cornering stiffness during a manoeuvre. The cornering stiffness of a tire is usually considered constant at low slip angles. However,
Karan Chatrath Master of Science Thesis 4-8 Additional Vehicle Dynamics Factors 55 the tire characteristics indicate that this is not the case for higher slip angles. Taking this variation into account results in the control effectiveness matrix becoming dynamic in nature (see equation 4-16). Each of these practical considerations is addressed as follows.
4-8-1 Tire Limits As Control Allocation Constraints
Here, the Dugoff tire model is considered. According to the equations laid out in section 2-3-2, the lateral force developed in each tire can be estimated. This lateral force can then be used to compute the maximum possible longitudinal force that the tire can develop. This is done using the concept of the friction circle (equation 2-17). Having computed the maximum possible longitudinal force that can be developed, the limiting brake torque can be found by multiplying that force with the effective radius of the wheel.
q 2 2 Fx,max = µFz − Fy (4-19)
Tmax = Fx,maxrw (4-20) The following few equations have been highlighted in chapter 3, they are however redescribed here for the sake of completeness. Looking back at the constraints on the brake actuators for the CA problem, we have uMin ≤ u(t) ≤ uMax (4-21)
ρMin ≤ u˙(t) ≤ ρMax (4-22) The rate limits can be re written as position limits by approximating the derivative as follows.
u (t) − u(t − T ) u˙ (t) ≈ (4-23) T Based on the actuator limits, the constraints are
ulower ≤ u ≤ uupper (4-24)
Where
ulower = max[uMin, u (t − T ) + T ρMin]
uupper = min[uMax, u (t − T ) + T ρMax] (4-25)
By incorporating the limits of longitudinal tire force generation (equation 4-20), the overall constraints are u ≤ u ≤ u (4-26) Where,
u = ulower
u = min(uupper,Tmax) (4-27)
Master of Science Thesis Karan Chatrath 56 Electronic Stability Control Using Control Allocation
4-8-2 Variation of Cornering Stiffness
The cornering stiffness of a tire is typically defined for a linear tire model as in equation 2-16. This quantity is taken as a constant assuming that the tire operates in a range of very small slip angles. However, the need for stability control arises when the vehicle is pushed to its limits of handling. The assumption of small slip angles in that region is not a suitable one. Moreover, using a linear tire model for investigations of such nonlinear manoeuvres is an oversimplification of the scenario. To understand the nature of variation of tire cornering stiffness, a case presented in the previous section is revisited. Specifically, case J (section 4-6) is looked at. In this simulation scenario, an extra step was carried out, which had no impact on the results of the simulation. This extra step involved understanding the variation of cornering stiffness as the manoeuvre progresses with time. Using the Dugoff model equation 2-18, we have Cαi tan αi Fyi = f (λ) (4-28) 1 − κi By assuming, small angles, one can write tan(α) = α. Dividing both sides by α leads to an expression Fy/α on the left-hand side of equation 4-28. This can be thought of as the cornering stiffness which varies with time and other tire-related quantities.
Each tire has a cornering stiffness associated with it. However, from a vehicle dynamics standpoint, it is useful to know the combined front cornering stiffness (used in the control effectiveness matrix). A plot of this time-varying front cornering stiffness can be studied in figure 4-9.
Figure 4-9 shows that the variation in the quantity is high and that treating it as a constant is not suitable. Based on this assessment, it was decided to dynamically update the front cornering stiffness in the control effectiveness matrix B. Upon inspecting the control effec- tiveness matrix a bit further, it was seen that the matrix has a few elements which are zero. This matrix has been shown in equation 4-29 again.
It was demonstrated earlier that the static control effectiveness matrix was derived by assum- ing small angles in the planar model. In this section, this assumption no longer holds true. The control effectiveness matrix is now derived using the planar model making a fewer num- ber of approximations. The linear tire model is still in use, however, the cornering stiffness is updated according to the Dugoff model, as described earlier.
− 1 − 1 − 1 − 1 0 rw rw rw rw B = 0 0 0 0 Cαf (4-29) T T f − f Tr − Tr C L 2rw 2rw 2rw 2rw αf f
This modification of the control effectiveness is inspired by the work carried out by Shyrokau [53]. The expression for the control effectiveness matrix is described in equation 4-30. This expression for the control effectiveness described in equation 4-30, the terms Cαfl and Cαfr are the front left and front right cornering stiffness values computed using the Dugoff tire model as described earlier.
Karan Chatrath Master of Science Thesis 4-8 Additional Vehicle Dynamics Factors 57
105 1.6
1.4
1.2
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0.6
0.4 8 9 10 11 12 13 14 15 16
Figure 4-9: Variation of Cornering stiffness with time during the sine with dwell manoeuvre
− cos(δ)/rw − cos(δ)/rw −1/rw −1/rw Bbrakes = − sin(δ)/rw − sin(δ)/rw 0 0 T cos(δ)−2L sin(δ) −T cos(δ)−2L sin(δ) f f f f Tr − Tr rw rw 2rw 2rw −(Cαfl + Cαfl) sin(δ) (C + C ) cos(δ) BAF S = αfl αfl (Cαfl−Cαfl)Tf sin(δ) (Cαfl + Cαfl) cos(δ) + 2 h i B = Bbrakes BAF S (4-30)
Master of Science Thesis Karan Chatrath 58 Electronic Stability Control Using Control Allocation
4-9 Simulations Scenarios: Cases K and M
Simulation Scenarios Cases K,L and M (section 4-6) take into consideration the vehicle dy- namics effects discussed in section 4-8. The APE+MPCA algorithm for this situation has a block diagram as described in figure 4-10.
Figure 4-10: APE with MPCA accounting for tire limits and dynamic control effectiveness
Figure 4-11: Sine With Dwell Test - Case K - Vehicle Response
Karan Chatrath Master of Science Thesis 4-9 Simulations Scenarios: Cases K and M 59
Figure 4-12: Sine With Dwell Test - Case K - Actuator Response
With this understanding, the simulation for scenario Case K was carried out and the results are presented in the figures 4-11 and 4-12.
Figure 4-12 can be compared with figure 4-7. It was mentioned earlier that integrated control results in lower brake torques being generated when compared to the use of only differential braking. This results in a lower loss of speed. It is seen that for a steering wheel angle am- plitude of 130 degrees, the vehicle speed reduces to approximately 62 km/hr when only four actuators are used, while the speed reduces to 65 km/hr when five actuators are used. This comparison can be done by looking at figures 4-11 and 4-6. The greater loss of speed when only using differential braking is probably insignificant to a driver. However, a larger loss of speed is expected in more aggressive manoeuvres, where higher brake torques are likely to be generated by the controller. The inference from this comparison is that integrated control is a better alternative to differential braking since it has a lower impact on other states of motion like longitudinal velocity.
The tuning parameters are as described in table 4-7. The difference between cases J and K is that case K incorporates tire limits and a dynamic control effectiveness matrix while case J does not. The results demonstrate that the vehicle is stabilised and the control allocation strategy is effective. Additional results for case K can be studied in AppendixB. A comparison for the results of cases J and K was carried out to see the benefit of incorporating additional vehicle dynamics factors. This comparison can be seen in figure 4-13. It can be seen that the yaw rate response is marginally better in case K as the vehicle stabilises faster than in
Master of Science Thesis Karan Chatrath 60 Electronic Stability Control Using Control Allocation case J. Also, the amount of body slip experienced by the vehicle in case K is lower. This is desirable since the vehicle experiences a lesser amount of unwanted lateral motion, or ’drift’.
Figure 4-13: Sine With Dwell Test - Case J and K Comparison - Vehicle Response - SWA - 130 degrees - Case J: Static B, Case K: Dyn B
4-9-1 Simulations With Hydraulic Brake Model
All simulations reported until case K consider simplified first-order brake actuator dynamics (section 2-5-1). However, to perform a more realistic assessment of the control system, the simple brake model has been replaced by a hydraulically actuated brake (HAB) model as described in section 2-5-1. This section reports observations made while simulating case M (refer section 4-6).
The control allocation technique in use in case M is APE+MPCA. An attempt is made to fit the response of the HAB actuator to a linear dynamical system. More specifically, a first-order transfer function. The results of case M can be studied in the following figures. In this case, only a SWA amplitude of 115 degrees is simulated. Simulations for more aggressive manoeuvres fail due to numerical issues.
It can be seen in figure 4-16 that the SWD test nearly fails. Moreover, in figure 4-14 it can be seen that the vehicle experiences body slip angles greater than 10 degrees and a comparatively greater loss of speed as it drops below 60 km/hr. Furthermore, the brake torques generated by the controller are of a very ’jerky’ nature as seen in figure 4-15. This kind of brake system response can cause passenger discomfort. Looking at the control allocation effectiveness in figure 4-17, it can be seen that the effectiveness considerably reduces.
Karan Chatrath Master of Science Thesis 4-9 Simulations Scenarios: Cases K and M 61
Figure 4-14: Sine With Dwell Test - Case M - Vehicle Response
Figure 4-15: Sine With Dwell Test - Case M - Vehicle Response
Master of Science Thesis Karan Chatrath 62 Electronic Stability Control Using Control Allocation
Figure 4-16: Sine With Dwell Test - Case M - SWA - 115 degrees
6
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Figure 4-17: Sine With Dwell Test - Case M - Control Allocation Effectiveness
This kind of result can be explained primarily by the nature of the hydraulic brake system in use. Referring to section 2-5-1, one can see that the HAB system is equipped with two sets of valves, each of which operates independently of each other. One set of valves is responsible for brake pressure build-up while the other regulates pressure release. On noticing figure 4-15, one can see that the rate at which pressure builds up is not the same as the rate at which it is
Karan Chatrath Master of Science Thesis 4-10 Summary 63 released. Different rates of increase and decrease cannot be fit into a linear first-order dynamic system which was the attempt made in case M. The nature of the HAB system dynamics is in fact, quite different from that predicted by a first-order transfer function. Approximating brake dynamics as a first order system has been carried out by [12]. In conclusion, by using simplified brake actuator dynamics, the APE+MPCA control allocation method shows promising results. However, the introduction of a complex HAB system renders the control allocation method to be ineffective.
4-10 Summary
In this chapter, the details of the ESC system design were presented. This was followed by an assessment of 13 simulation scenarios starting named as case A to M (refer section 4-6). Certain key results have been presented in this chapter while most of the results are presented in appendixB. It was observed that the APE+MPCA control allocation technique was ef- fective when the brake actuator dynamics were described by a first-order transfer function. Moreover, it was observed that the response of the Pfeffer Steering model can be fit to a first-order transfer function and doing so resulted in no loss of control allocation effectiveness. On introducing the more realistic HAB actuator model, it was seen that the APE+MPCA technique loses effectiveness. The response of the brake system cannot be fitted to a first- order linear system since the rate at which brake pressures (or torque) builds up and the rate at which the system releases pressure is different. Modelling such a system requires a more detailed and possibly non-linear mathematical description.
Master of Science Thesis Karan Chatrath 64 Electronic Stability Control Using Control Allocation
Karan Chatrath Master of Science Thesis Chapter 5
Conclusions And Recommendations
5-1 Highlights And Conclusions
The primary goal of this thesis was to investigate the concept of control allocation and apply the technique for vehicle stability control. It was found that the Control Allocation (CA) strategy enables optimal distribution of high-level control commands among available actua- tors. Furthermore,CA can perform such an optimal distribution by accounting for the limits of the actuators.
To accomplish the design of electronic stability control systems using control allocation, two actuator configurations were of focus. One configuration comprised of four actuators (differ- ential braking) while the second comprised of five actuators (Differential braking and active front steering). For each of these actuator configurations, the Weighted Least Squares (WLS) CA method was at first applied without considering actuator dynamics. The results obtained for each of these cases demonstrated the benefit of coordinating a larger number of actuators. The high-level yaw moment correction when distributed among a larger number of actuators resulted in the generation of lower brake torques by the control system. This, in turn, resulted in a lower unwanted loss of speed at the end of the sine with dwell manoeuvre. In conclusion, the use of integrated control results in lower actuator commands and better vehicle perfor- mance.
Upon the introduction of actuator dynamics (governed by transfer functions), a deterioration of the effectiveness of the WLS method was visible. This deterioration manifests itself in two ways. The actuators fail to meet the demand generated by the high-level controller and the closed-loop system performance deteriorates. The need to account for actuator dynamics was established and this warranted the use of Model Predictive Control Allocation (MPCA). It was seen that the MPCA method overcomes the shortcomings of the WLS and other such methods that do not consider actuator dynamics.
Master of Science Thesis Karan Chatrath 66 Conclusions And Recommendations
This work also drew attention to the fact that MPCA loses effectiveness when uncertainties in actuator dynamics are present. This was addressed by combining the MPC allocator along with an adaptive parameter estimator. It was shown that the APE+MPCA control allocation technique successfully handles actuator uncertainties when the actuators obey linear dynam- ics.
In reality, actuator responses are more complex than what simple linear dynamics can predict. With this understanding, keeping the brake actuators simple, the linear steering actuator dy- namic system was replaced with a more detailed Pfeffer Steering Model. An attempt was made to fit the response of the steering system to a linear dynamical system by using input- output data. The results demonstrated that the APE+MPCA strategy performs effectively despite the introduction of a complex steering model.
Finally, the linear brake system actuators were replaced with a detailed Hydraulic brake model. Upon attempting to fit the response of the brake system to a linear dynamic system, it was seen that the APE+MPCA strategy loses effectiveness. The reason for this result is that the brake system comprises two sets of valves that operate independently. This resulted in rates of brake pressure build-up and release to be different. Such behaviour cannot be modelled or fit as a linear dynamic system. It can be concluded that simplified actuator models for brakes inadequately capture hydraulic brake system behaviour.
In conclusion, the APE+MPCA control allocation technique can effectively deal with actuator uncertainties as long as the actuators are governed by uncertain linear dynamics. The method also shows promise in some situations where the actuators in use are unknown and nonlinear. In this case, theCA solver successfully handles complex steering dynamics effectively. The APE+MPCA technique loses effectiveness when actuator behaviours are too complex to be fit to a linear dynamical system.
5-2 Recommendations And Scope For Future Work
The author draws attention to the first sentence of the first chapter of this report. ’Ad- vancement is a continuous process’. The scope for improvement in any field is never-ending. Recognizing and acting on this scope is the key to advancement. The field of vehicle dynam- ics control is no exception to this fact. Automotive companies’ research and development activities are constantly working towards the betterment of Advanced Driver Assistance Sys- tems (ADAS) and automated driving. Considering all the work carried out in this thesis, the following recommendations are made:
• The high-level controller used in this thesis is a simple discrete-time PD controller. This has a limited range of operation as seen in the simulations. The vehicle, for most cases, could not be controlled for more aggressive sine with dwell manoeuvres (>130 degrees of SWA). Use of techniques such as feedback linearization or even high-level nonlinear
Karan Chatrath Master of Science Thesis 5-2 Recommendations And Scope For Future Work 67
model predictive control could prove to be a better choice than a simple PD compen- sator.
• In the APE+MPCA strategy, the adaptive parameter estimator is independent of the MPCA solver. In this case, only the identified parameters are fed into the MPCA solver. The parameter estimation technique used in this work is the AM - RLS technique which operates under the assumptions that the order and structure of the dynamical system is known and is linear, the input delay (if any) is also a known quantity. The AM-RLS technique can be easily replaced with a parameter estimator that makes fewer assump- tions. It is possible that the linear MPCA solver combined with a better parameter estimator can result in more effective control allocation.
• The actuator dynamics used for the most part of this thesis are linear in nature. It was demonstrated that by simulating the control system with more realistic brake actuator descriptions, the control allocation effectiveness reduces. It is therefore highly recom- mended to carry out investigations with realistic actuator descriptions as opposed to using simplified models. If the governing dynamics of the actuators become nonlinear, then the MPCA solver would also be of a nonlinear nature. The use of nonlinear MPCA is expected to handle nonlinear actuator dynamics effectively. Also, actuator aging can be a potential problem. Nonlinear MPCA combined with nonlinear parameter estima- tors can result in fault-tolerant control allocation.
• From a vehicle dynamics standpoint, the controlled vehicle has been subjected to the standard ISO sine with dwell test. The controller can be evaluated in different manoeu- vres in both simulations as well as real-time testing. More specifically, scenarios, where the vehicle is subjected to unknown disturbances, low friction conditions, poor road conditions must be looked into to assess the robustness of the overall control system.
Master of Science Thesis Karan Chatrath 68 Conclusions And Recommendations
Karan Chatrath Master of Science Thesis Appendix A
Planar vehicle model and Dugoff model validation
Parameter Value Total Mass - m 1963 kg Distance - front front axle to COG - Lf 1.0935 m Diatance - rear axle to COG - Lr 1.569 m Front track width - Tf 1.616 m Rear track width - Tr 1.613 m Height of COG - hcg 0.673 m Effective wheel radius - rw 0.3706 m Friction Coefficient - µ 1 Front axle cornering stiffness - Cαf 148668 N/rad Rear axle cornering stiffness - Cαr 166850 N/rad 2 Moment of Inertia about vehicle Z axis - Izz 3386.095 kg - m Steering Ratio - Sratio 16
Table A-1: Vehicle model parameters - Non exhaustive list - From IPG CarMaker
Master of Science Thesis Karan Chatrath 70 Planar vehicle model and Dugoff model validation
Figure A-1: Validation with Steering wheel angle amplitude of 100 degrees
Figure A-2: Validation with Steering wheel angle amplitude of 110 degrees
Karan Chatrath Master of Science Thesis 71
Figure A-3: Validation with Steering wheel angle amplitude of 130 degrees
Figure A-4: Dugoff Tire Model Validation - SWA - 130 degrees - TM (Tire Model)
Master of Science Thesis Karan Chatrath 72 Planar vehicle model and Dugoff model validation
Karan Chatrath Master of Science Thesis Appendix B
ESC with Control Allocation - Results
The specifics of each following case can be found in section 4-6.
B-1 Case A Results
Figure B-1: Sine With Dwell - Case A - Vehicle Response
Master of Science Thesis Karan Chatrath 74 ESC with Control Allocation - Results
Figure B-2: Sine With Dwell - Case A - Actuator Response - Brake Torques
Figure B-3: Sine With Dwell - Case A - SWA - 130 degrees
Karan Chatrath Master of Science Thesis B-2 Case B Results 75
2.5
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Figure B-4: Sine With Dwell - Case A - SWA - 130 degrees - Control Allocation Effectiveness
B-2 Case B Results
Figure B-5: Sine With Dwell - Case A and B comparison - Vehicle Response Legend: NoDyn - Case A, ActDyn - Case B
Master of Science Thesis Karan Chatrath 76 ESC with Control Allocation - Results
Figure B-6: Sine With Dwell - Case A and B comparison - Actuator Response - Brake Torques - Legend: NoDyn - Case A, ActDyn - Case B
Figure B-7: Sine With Dwell - Case B - SWA - 130 degrees
Karan Chatrath Master of Science Thesis B-3 Case C Results 77
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Figure B-8: Sine With Dwell - Case B - SWA - 130 degrees - Control Allocation Effectiveness
B-3 Case C Results
Figure B-9: Sine With Dwell - Case C - Vehicle Response
Master of Science Thesis Karan Chatrath 78 ESC with Control Allocation - Results
Figure B-10: Sine With Dwell - Case C - Actuator Response - Brake Torques
Figure B-11: Sine With Dwell - Case C - SWA - 130 degrees
Karan Chatrath Master of Science Thesis B-4 Case D Results 79
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Figure B-12: Sine With Dwell - Case C - SWA - 130 degrees - Control Allocation Effectiveness
B-4 Case D Results
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Figure B-13: Sine With Dwell - Case D - SWA - 130 degrees - Control Allocation Effectiveness
Master of Science Thesis Karan Chatrath 80 ESC with Control Allocation - Results
Figure B-14: Sine With Dwell - Case D - SWA - 130 degrees
B-5 Case E Results
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Figure B-15: Sine With Dwell - Case E - SWA - 130 degrees - Control Allocation Effectiveness
Karan Chatrath Master of Science Thesis B-6 Case F Results 81
Figure B-16: Sine With Dwell - Case E - SWA - 130 degrees - Brake Actuators’ parameter estimation convergence
B-6 Case F Results
Figure B-17: Sine With Dwell - Case F - Vehicle Response
Master of Science Thesis Karan Chatrath 82 ESC with Control Allocation - Results
Figure B-18: Sine With Dwell - Case F - Actuator Response
Figure B-19: Sine With Dwell - Case F - SWA - 130 degrees
Karan Chatrath Master of Science Thesis B-7 Case G Results 83
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Figure B-20: Sine With Dwell - Case F - SWA - 130 degrees - Control Allocation Effectiveness
B-7 Case G Results
Figure B-21: Sine With Dwell - Case F and G comparison - Vehicle Response - Legend: NoDyn - Case F, ActDyn - Case G
Master of Science Thesis Karan Chatrath 84 ESC with Control Allocation - Results
Figure B-22: Sine With Dwell - Case F and G comparison - Actuator Response - Legend: NoDyn - Case F, ActDyn - Case G
Figure B-23: Sine With Dwell - Case G - SWA - 130 degrees
Karan Chatrath Master of Science Thesis B-8 Case H Results 85
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Figure B-24: Sine With Dwell - Case G - SWA - 130 degrees - Control Allocation Effectiveness
B-8 Case H Results
Figure B-25: Sine With Dwell - Case H - Vehicle Response
Master of Science Thesis Karan Chatrath 86 ESC with Control Allocation - Results
Figure B-26: Sine With Dwell - Case H - Actuator Response
Figure B-27: Sine With Dwell - Case H - SWA - 130 degrees
Karan Chatrath Master of Science Thesis B-9 Case I Results 87
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Figure B-28: Sine With Dwell - Case H - SWA - 130 degrees - Control Allocation Effectiveness
B-9 Case I Results
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Figure B-29: Sine With Dwell - Case I - SWA - 130 degrees - Control Allocation Effectiveness
Master of Science Thesis Karan Chatrath 88 ESC with Control Allocation - Results
Figure B-30: Sine With Dwell - Case I - SWA - 130 degrees
B-10 Case J Results
Figure B-31: Sine With Dwell - Case J - Vehicle Response
Karan Chatrath Master of Science Thesis B-10 Case J Results 89
Figure B-32: Sine With Dwell - Case J - Actuator Response - Brake Torques
Figure B-33: Sine With Dwell - Case J - SWA - 130 degrees
Master of Science Thesis Karan Chatrath 90 ESC with Control Allocation - Results
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Figure B-34: Sine With Dwell - Case J - SWA - 130 degrees - Control Allocation Effectiveness
Figure B-35: Sine With Dwell - Case J - Brake Actuator Parameter Estimation
Karan Chatrath Master of Science Thesis B-11 Case K Results 91
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Figure B-36: Sine With Dwell - Case J - Pfeffer Steering Actuator Parameter Estimation
B-11 Case K Results
Figure B-37: Sine With Dwell - Case K - SWA - 130 degrees
Master of Science Thesis Karan Chatrath 92 ESC with Control Allocation - Results
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Figure B-38: Sine With Dwell - Case K - SWA - 130 degrees - Control Allocation Effectiveness
Figure B-39: Sine With Dwell - Case K - Brake Actuator Parameter Estimation
Karan Chatrath Master of Science Thesis B-12 Case L Results 93
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Figure B-40: Sine With Dwell - Case K - Pfeffer Steering Actuator Parameter Estimation
B-12 Case L Results
Figure B-41: Sine With Dwell - Case L - Vehicle Response
Master of Science Thesis Karan Chatrath 94 ESC with Control Allocation - Results
Figure B-42: Sine With Dwell - Case L - Actuator Response - Brake Torques
Figure B-43: Sine With Dwell - Case L - SWA - 130 degrees
Karan Chatrath Master of Science Thesis B-13 Case M Results 95
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Figure B-44: Sine With Dwell - Case L - SWA - 130 degrees - Control Allocation Effectiveness
B-13 Case M Results
Figure B-45: Sine With Dwell - Case M - Brake Actuator Parameter Estimation - SWA - 115 degrees
Master of Science Thesis Karan Chatrath 96 ESC with Control Allocation - Results
Karan Chatrath Master of Science Thesis Bibliography
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Karan Chatrath Master of Science Thesis Glossary
List of Acronyms
CA Control Allocation
ESC Electronic Stability Control
MPC Model Predictive Control
SRS Supplementary Restraint Systems
ADAS Advanced Driver Assistance Systems
VDC Vehicle Dynamics Control
VSC Vehicle Stability Control
ABS Antilock Braking System
TCS Traction Control System
DOF Degree/s Of Freedom
MPCA Model Predictive Control Allocation
APE Adaptive Parameter Estimation
WLS Weighted Least Squares
ESP Electronic Stability Program
COG Center Of Gravity
SLS Sequential Least Squares
MLS Minimal Least Squares
CGI Cascaded Generalised Inverses
AM-RLS Auxillary Model Based Recursive Least Squares
Master of Science Thesis Karan Chatrath 102 Glossary
APE Adaptive Parameter Estimation
SWA Steering Wheel Angle/ Steering Wheel Angle Amplitude
HAB Hydraulically Actuated Brake Model
AFS Active Front Steering
EPS Electronic Power Steering
SWD Sine With Dwell
ISO International Standards Organisation
QCAT Quadratic Programming Control Allocation Toolbox
ZOH Zero Order Hold
Karan Chatrath Master of Science Thesis