Enhanced Sliding Mode Controls for Autonomous Vehicles

Treball de Fi de Master Master’s Degree in Automatic Control and Robotics (MUAR) Enhanced Sliding Mode Controls for Autonomous Vehicles Author: Daniel Ye Director: Dr. Vicenç Puig Cayuela Announcement: Juny 2024 Higher Technical School of Industrial Engineering of Barcelona 3 Summary Aquest treball presenta una solució per al control de la direcció i l'acceleració dels vehicles autònoms. Es desenvolupen dos models paramètrics basats en la física, cinemàtic i dinàmic, per a vehicles de quatre rodes amb tracció posterior. Es realitza una estratègia de control no lineal utilitzant el sliding mode control. S'apliquen tres estratègies de tuning, pseudogreedy search, algorismes genètics i particles swarm optimization, per determinar els paràmetres dels controladors. S'estudia una comparació entre el sliding mode control i el linear paramter varying model predictive control. La robustesa contra la no linealitat i la incertesa paramètrica en el model són els principals avantatges del sliding mode control. No obstant això, el tremolor és el principal inconvenient; per tant, es implementa un disseny avançat en el sliding surface. Les estratègies de control i les estratègies de tuning proposades s'implementen amb Matlab en un entorn virtual al núvol. A més, es presenta el desenvolupament teòric del linear matrix inequality-based sliding mode control utilitzant un model dinàmic considerant l'efecte de l'angle de lliscament. 4 Abstract This paper presents a solution to the steering and acceleration control of autonomous vehicles. Two parametric physics-based models, kinematic and dynamic, are developed for four-wheeled rear-wheel drive vehicles. A nonlinear control strategy is performed using sliding mode control. Three tuning strategies, pseudo-greedy search, genetic algorithms, and particle swarm optimization are applied to determine controller parameters. A comparison of sliding mode control and linear parameter-varying model predictive control is studied. Robustness against nonlinearity and parametric uncertainty in the model are the main advantages of the sliding control method. Nevertheless, chattering is the main drawback; thus, advanced design in the sliding surface is implemented. The proposed control strategies and tuning strategies are implemented with Matlab in a cloud virtual environment. Additionally, the theoretical development of linear matrix inequality-based sliding mode control using a dynamic model considering the effect of slip angle is presented. 5 Acknowledgements After completing this work, I would like to give thanks to several people who helped and given me supports. Firstly, I want to thank the ETSEIB, to the professor for teaching me the professional knowledge of engineering and lead me more into the world of robotic and automatic control. Specially thank to my coordinator Puig Vicenç, professor in Automatic Control at Institut de Robòtica i Informàtica Industrial (IRI), who illustrated me the topic of this work, and for his detailed and patient guidance during the research and development. Secondly, I also want to thank my friends and family. They are the ones who give me emotional and moral support and the support in daily life. Moreover, they taught me the from their experience the attitude towards life and work. Finally, I want to thank all the antecedent researcher and worker in the automatic control, robotic and autonomous vehicles field, whether mathematician like Lyapunov or specialist from other implicitly related field or author of project and software, without them and their works and resource, I wouldn’t be able to perform this work. Thank you, Barcelona, Juny 2022 Daniel Ye 7 Content Summary 3 Abstract 4 Acknowledgements 5 Content 7 Glossary 9 List of figures 10 List of tables 11 1. Introduction 13 1.1. Motivation................................................................................. 13 1.2. Prerequisites .............................................................................. 13 1.3. Objectives ................................................................................. 14 2. Theoretical Background 15 2.1. Autonomous Vehicles.................................................................. 15 2.2. Previews works .......................................................................... 16 2.3. Control Theory .......................................................................... 17 3. Model Description 25 3.1. Kinematic Model ........................................................................ 25 3.2. Dynamic Model .......................................................................... 27 3.3. Nomenclature ............................................................................ 30 4. Control Strategies 31 4.1. Nonlinear Sliding Mode Control ................................................... 31 4.2. Linear Parameter Varying Model Predictive Control ...................... 33 5. Implementation 35 5.1. Simulation Environment ............................................................. 35 8 5.2. Tuning Strategy ......................................................................... 39 6. Simulation 46 6.1. Result Overview ......................................................................... 46 6.2. NLSMC..................................................................................... 47 6.3. LPVMPC .................................................................................. 49 6.4. Further simulation ..................................................................... 51 6.5. Comparison ............................................................................... 53 7. Planning 54 8. Project Budget 55 9. Economy, environment, society, and equality 56 9.1. Economy ................................................................................... 56 9.2. Environment .............................................................................. 56 9.3. Society ...................................................................................... 57 9.4. Equality .................................................................................... 57 10. Future Work 58 10.1. LPV-LQR-SMC ......................................................................... 58 10.2. Slip Angle.................................................................................. 60 CONCLUSIONS 61 References 62 Annex 65 Annex A. Complete plot of NLSMC ........................................................ 65 Annex B. Complete plot of LPVSMC ...................................................... 68 9 Glossary AV: Autonomous Vehicles EV: Electrical Vehicles GA: Genetic Algorithm LMI: Linear Matrix Inequality LPV: Linear Parameter Varying LQR: Linear Quadratic Regulator MPC: Model Predictive Control MT: Manual Tuning NHTSA: National Highway Traffic Safety Administration NL: Non Linear pGS: pseudo-Greedy Search PSO: Particle Swarm Optimization RSME: Root Square Mean Error SF: Sliding Surface SMC: Sliding Mode Control TFM: Treball Final de Master (Final Master Work) 16 Enhanced Sliding Mode Controls for Autonomous Vehicles 2.2. Previews works Table 2.1 Comparative summary of controllers and articles of autonomous vehicles [5]. Control Method Model Type Advantage Drawback Related Work MPC Kinematic Dynamic Linear Nonlinear Optimal results Handles constraints Predictive feature MIMO systems Online optimization High computation cost Need accurate model [6] [7] [8] [9] SMC Kinematic Dynamic Linear Nonlinear Strong robustness Nonlinear systems MIMO systems Switching systems Low computation cost Chattering phenomenon Non continuous control Cannot add constraints [10] [11] LQR Kinematic Dynamic Linear Optimal results MIMO systems Off-line optimization Easy to implement Linearization assumptions Need states measurable Need accurate model Poor robustness [12] [13] LPV Geometric Kinematic Dynamic Linear Capture nonlinearity Feedback control Switching systems Low computation cost Difficult design Requires instantiation Requires accurate model Limited stability [8] [14] [15] Chapter 2. Theoretical Background 17 2.3. Control Theory The core idea of the control theory is to automatically generate a sequence control signal based on the real time feedback signal, which normally is the error of the system output respect to the reference. The control signal is applied to the dynamic system with objective of stability and performance. This type is called closed loop negative feedback control. 2.3.1. State Space The state space is the mathematical representation of physical system, composed by the input (u), output (y), variables (x) and featured by the differential equations. 𝑥󰇗=𝑑𝑥 𝑑𝑡=𝑓(𝑡,𝑥,𝑢) (2.1𝑎) 𝑦=ℎ(𝑡,𝑥,𝑢) (2.1𝑏) 𝑥∈ℝ𝑛 𝑢∈ℝ𝑛𝑢 𝑦∈ℝ𝑚 For the linear or linearized system, the state space can be written as matrix 𝑥󰇗=𝐴𝑥+𝐵𝑢 (2.2𝑎) 𝑦=𝐶𝑥+𝐷𝑢 (2.2𝑏) In the real application, since the instrumentation is limited by the sampling time (Ts) with is not infinitely small, then the state space also should be discretized, below show an example of discretization by using Euler approximation. 𝑑𝑥 𝑑𝑡≅𝑥(𝑘+1)−𝑥(𝑘) 𝑇𝑠,𝑘=1,2,…,𝑁 (2.3) 𝑥(𝑘+1)=𝐴𝑑𝑥(𝑘)+𝐵𝑑𝑢(𝑘) (2.4𝑎) 𝑦(𝑘)=𝐶𝑑𝑥(𝑘)+𝐷𝑑𝑢(𝑘) (2.4𝑏) Where, 𝐴𝑑=𝑒𝐴𝑇𝑠 𝐵𝑑=(𝐴𝑑−𝐼)𝐴−1𝐵 (2.4𝑐) 𝐶𝑑=𝐶 𝐷𝑑=𝐷 (2.4𝑑) Notice the matrix A should be symmetric to guarantee the inversibility. The output signal (y) will be measured by the sensor then used to error computations [16]. 18 Enhanced Sliding Mode Controls for Autonomous Vehicles 2.3.2. Linearization Nonlinear system can be linearized around the equilibrium point, where the derivatives of the states are equal to zero. 0=𝑓(𝑥∗,𝑢∗) (2.5𝑎) 𝑦=ℎ(𝑥∗,𝑢∗) (2.5𝑏) The linearized system is obtained with Taylor expansion of one order, which is the partial differentiation of the original system and substitution with the value of state and input at the equilibrium point. 𝐴≜𝜕𝑓 𝜕𝑥|𝑥∗,𝑢∗ 𝐵≜𝜕𝑓 𝜕𝑢|𝑥∗,𝑢∗(2.6𝑎) 𝐶≜𝜕ℎ 𝜕𝑥|𝑥∗,𝑢∗ 𝐷≜𝜕ℎ 𝜕𝑢|𝑥∗,𝑢∗(2.6𝑏) Due to the approximation is made around the point, the linearized system shows high error when the state is far to the equilibrium point. The linearization of system is highly interested because the main part control theory and technique is obtained from linear system. Furthermore, the nonlinear system has not systematic manner to study due to it chaotic behaviors [16]. 2.3.3. Optimal Control The optimal control theory is a branch of mathematical optimization, the aim of which is to find the control signal for the dynamic system in a period optimizing the cost function J. minimize 𝐽(𝑥,𝑢) 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡𝑠 (2.7) There exits to type of optimality, the global and local. The former is the real optimal result, and the latter is the result that optimizer solver will fall into due to capacity of the solver. 2.3.4. State Feedback Control The state feedback control, also known as pole placement, is the most intuitive and classical control, the control law is defined as the state multiplied by a negative constant. 𝑢=−𝐾𝑥 (2.8) This controller is intuitive because it leads the system to decrease when the current state is higher than the desired reference, and to increase in the opposite situation. Chapter 2. Theoretical Background 19 2.3.5. Linear Quadratic Regulator The LQR is a feedback controller with quadratic cost function 𝐽=𝑥𝑇𝑆𝑥+∫ (𝑥𝑇𝑄𝑥+𝑢𝑇𝑅𝑢+2𝑥𝑇𝑁𝑢)𝑑𝑥 𝑡1 𝑡0(2.9) Where the S, Q, R and N are diagonal matrices of dimensions concordant to the multiplied state or input. They represent the weight of each state and each input. The physical meaning of this cost function is that the state (or error for the reference tracking case) and control signal is optimized to minimum, therefore, the tracking error and the energy consumption is reduced. The main drawback of the LQR is quantity of work to specify the value of parameters S, Q, R and N to obtain the best performance (tunning) [17]. 2.3.6. Lyapunov Stability Theory For a nonlinear autonomous (time invariant) system 𝑑𝑥 𝑑𝑡=𝑓(𝑥) (2.10) If we define a Lyapunov function V(x) in manner fulfilling: 1. 𝑉(𝑥)=0 𝑎𝑡 𝑥=0 2. positive definite: 𝑉(𝑥)>0,∀𝑥≠0 3. negative definite: 𝑉󰇗(𝑥)<0,∀𝑥≠0 4. radially unbounded: 𝑉(𝑥)→∞ 𝑎𝑠 ‖𝑥‖→∞ Then the system is globally asymptotically stable (stable in Lyapunov Theory sense). The selection of V(x) is a great subject to study, is rather common to select the quadratic form of the states (x). For a case of linear or linearized system 𝑥󰇗=𝐴𝑥 (2.11) The selection of the Lyapunov function is 𝑉(𝑥)=𝑥𝑇𝑃𝑥 (2.12) The matrix P should be symmetric and positive definite to ensure therefore the V. 20 Enhanced Sliding Mode Controls for Autonomous Vehicles Then the derivative is 𝑉󰇗(𝑥)=𝑥󰇗𝑇𝑃𝑥+𝑥𝑇𝑃𝑥󰇗 =𝑥𝑇𝐴𝑇𝑃𝑥+𝑥𝑇𝑃𝐴𝑥 =𝑥𝑇(𝐴𝑇𝑃+𝑃𝐴)𝑥(2.13) To be negative definite the condition is 𝐴𝑇𝑃+𝑃𝐴<0 (2.14) The matrix P is found with numerical solver [18]. 2.3.7. Linear Matrix Inequality The linear matrix inequality [19] consists of to formulate or rewrite the optimization problem in manner the constraints are inequality expression independent of the desired term, which is more appropriate for the numerical solver to find the solution. Giving example of state feedback for linear system 𝑥󰇗=𝐴𝑥+𝐵𝑢=𝐴𝑥+𝐵(−𝐾𝑥)=(𝐴−𝐵𝐾)𝑥(2.15) Using Lyapunov function 𝑉(𝑥)=𝑥𝑇𝑃𝑥 (2.16) Then as same procedure of (2.13), the condition to fulfill become (𝐴−𝐵𝐾)𝑇𝑃+𝑃(𝐴−𝐵𝐾)<0 (2.17𝑎) Expanding the expression and applying change of variable, the final LMI is 𝐴𝑃+𝑃𝐴𝑇+𝐵𝑊+𝑊𝑇𝐵𝑇<0 (2.17𝑏) 𝑃>0 (2.17𝑐) Where 𝑊=−𝐾𝑃 (2.17𝑑) The matrix W and P are both unknow which should be found in the numerical computation, once the solution is found, the controller gain matrix K is calculated as 𝐾=−𝑊𝑃−1 (2.17𝑒) Notice this LMI only uses constraints and has not cost function, hence it only guarantees the stability of the system and nothing more. Chapter 2. Theoretical Background 21 2.3.8. Linear Parameter Varying When we have a moving system, the number of equilibrium point is infinite which lead to infinite linearization and optimization. A feasible method to handle this type of system is to use LPV method [15] [20], therefore, the fundamental ideal is to convert the nonlinear system to a finite number of linear systems. The LPV system is expressed as, 𝑥󰇗=𝐴(ϑ)𝑥+𝐵(ϑ)𝑢(2.18𝑎) 𝑦=𝐶(ϑ)𝑥+𝐷(ϑ)𝑢(2.18𝑏) 𝜃∈ℝ𝑛𝑠(2.18𝑐) Notice the matrix A, B, C and D are not fully numeric like linear system but depending on the scheduling variables ϑ, with 𝑛𝑠 number of elements. With known bound value (maximum and minimum) of all scheduling variables, we define the system in the extreme cases, having in total 𝑁=2𝑛𝑠 number of subsystems, also called vertex systems. Depending on the current value of the scheduling variables, the current system is computed by the interpolation of the subsystems. The polytopic LPV model is. 𝑥󰇗=∑𝜇𝑖(ϑ)(𝐴𝑖𝑥+𝐵𝑖𝑢) 𝑁 𝑖=1 (2.19𝑎) 𝑦=∑𝜇𝑖(ϑ)(𝐶𝑖𝑥+𝐷𝑖𝑢) 𝑁 𝑖=1 (2.19𝑏) The 𝜇𝑖 is the coefficient of the polytopic decomposition in the range [0, 1]. ∑𝜇𝑖 𝑁 𝑖=1 =1 (2.20𝑎) 𝜇𝑖(ϑ)=∏𝜉𝑖𝑗(𝜂𝑗𝑚𝑖𝑛 ,𝜂𝑗𝑚𝑎𝑥) 𝑛𝑠 𝑗=1 (2.20𝑏) Where the 𝜂𝑖 is the weight of the current scheduling variable obtained by the interpolation. 𝜂𝑗𝑚𝑖𝑛 =ϑ𝑗𝑚𝑎𝑥 −ϑ𝑗 ϑ𝑗𝑚𝑎𝑥−ϑ𝑗𝑚𝑖𝑛 (2.20𝑐) 𝜂𝑗𝑚𝑎𝑥 =1−𝜂𝑗𝑚𝑖𝑛 (2.20𝑑) 22 Enhanced Sliding Mode Controls for Autonomous Vehicles The state feedback controller is also converted to be weighted. 𝑢=−∑𝜇𝑖(ϑ)𝐾𝑖𝑥 𝑁 𝑖=1 (2.21) Notice the LPV-LMI optimization problem is not solved for each vertex separately; all the constraints are added at same time and sharing the unique matrix P. 2.3.9. Sliding Mode Control The error of the state be 𝑥𝑒=𝑥−𝑥𝑟𝑒𝑓 (2.22) The sliding surface or sliding variable has form 𝑠(𝑥𝑒,𝑡)=(𝑑 𝑑𝑡+𝜆)𝑛−1𝑥𝑒(2.23) For n = 2 𝑠=𝑥󰇗𝑒+𝜆𝑥𝑒(2.24) For n = 3 𝑠=𝑥󰇘𝑒+2𝜆𝑥󰇗𝑒+𝜆2𝑥𝑒(2.25) The sense to the sliding surface definition is that when the system reaches to the s = 0 the system enhance behavior 𝑥󰇗𝑒=−𝜆𝑥𝑒(2.26) Which is state feedback and ensure the system error converge to zero. The stability of the sliding mode control is proved with Lyapunov theory, let Lyapunov function be 𝑉(𝑠)=12𝑠𝑇𝑠>0 (2.27) The ideal performance in the sliding surface s=0 is 𝑠󰇗=−𝑞𝑠−𝑝sgn(𝑠) (2.28) Where the parameter q and p are positive constant to be tuned, the proportional term -qs can force the system reach to the sliding surface when s is large. The sgn() function defined as sgn(𝑠)={−1 𝑠<0 0 𝑠=0 1 𝑠>0 (2.29) Chapter 2. Theoretical Background 23 The derivative of the (2.27) is 𝑉󰇗(𝑠)=𝑠𝑠󰇗=−𝑞𝑠2+𝑠(−𝑝sgn(𝑠)) (2.30) The former term of (2.30) is ensured to be zero since its negative of a square, and for the latter term 𝐼𝑓 𝑠>0⟹sgn(𝑠)>0⟹−𝑝sgn(𝑠)<0 (2.31𝑎) 𝐼𝑓 𝑠<0⟹sgn(𝑠)<0⟹−𝑝sgn(𝑠)>0 (2.31𝑏) Hence, the (2.30) is guaranteed to be negative definite and the system fulfill the Lyapunov stability conditions. The graphical interpretation of the (2.31) is shown in the Figure 2.1. Where once reached the sliding surface, the system will converge exponentially with slope of 1/λ. Figure 2.1. Graphical interpretation of sliding mode control. The main drawback of the nonlinear SMC is the presence of the ‘chattering’ in the resulting control signal. Figure 2.2. Chattering. The chattering can damage the physical controller of the real system, it can be avoided by applying smoother function instead of the sgn(), for instance the saturation function 24 Enhanced Sliding Mode Controls for Autonomous Vehicles sat(𝑠)={−1 𝑠<1 𝑠 −1≤𝑠≤1 1 𝑠>1 (2.32) And advanced bounding saturation can be applied to filter better sat(𝑠𝜙)= { −1 𝑠𝜙<1 𝑠𝜙−1≤𝑠𝜙≤1 1 𝑠𝜙>1 (2.33) The drawback of (2.33) is that additional parameter 𝜙 are added and will take more cost in the tuning [21]. 2.3.10. Model Predictive Control The model predictive control [22] strategy differentiates from the classical in the prediction behavior. The classical control method preforms one time optimization, then the resulting control signal is deterministic along the simulation. Hence, the classical control method is unable to act when unexpected situation occurs during the simulation. Model predictive control, in the contrary, preform an optimization in each iteration, where prediction (simulation) is made until the specified prediction horizon (Hp) for each step of the system. Generating a sequence of predicted control signal from the current step to the Hp, where only the first item is applied to the system. The MPC formulation in discrete time is minimize 𝐽(𝑥,∆𝑢) 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 𝑥(𝑘+𝑖+1)=𝑓(𝑥(𝑘+𝑖),𝑢(𝑘+𝑖)) 𝑦(𝑘+𝑖)=𝑔(𝑥(𝑘+𝑖),𝑢(𝑘+𝑖)) 𝑖=0,1,…,𝐻𝑝−1 𝑢𝑚𝑖𝑛≤𝑢(𝑘+𝑖)≤𝑢𝑚𝑎𝑥 𝑖=0,1,…,𝐻𝑝−1 𝑥𝑚𝑖𝑛≤𝑥(𝑘+𝑖)≤𝑥𝑚𝑎𝑥 𝑖=1,2,…,𝐻𝑝 (2.34) The advantages of the MPC, a part of predictive behavior, since it is formulated in LMI form, it can be combined with other techniques like LQR to guarantee the stability and performance. Also, more state can be easily added to the system model without changing the MPC structure but increasing the state and control vector length. High computation cost for large prediction horizon and possible infeasible solution due to complexity of the formulated problem are the main drawbacks of the MPC. Chapter 3. Model Description 25 3. Model Description There exists multiple manner to model a real system, the Dynamic System Modeling (DSM) is the study of this topic. In this section, the modeling the autonomous vehicle based on the physic equation from kinematic and dynamic behaviors. 3.1. Kinematic Model Four-wheel car with rear-wheel drive (RWD) can be simplified as two-wheel RWD, which is the same structure of a bicycle [23]. Figure 3.1. Geometry of a simplified bicycle model of a four wheel vehicle. The kinematic model describes the motion of the vehicle respect the fixed world coordinate frame. Using simple trigonometry theorems, we obtain 𝑥󰇗=𝑣sin(𝜃) (3.1𝑎) 𝑦󰇗=𝑣cos(𝜃) (3.1𝑏) 𝜃󰇗=𝑣𝑙𝑟tan(𝛿) (3.1𝑐) Where the v is the speed of the vehicle, the x, y and θ is the position and orientation, the δ is the steering angle of the steering wheel. The 𝑙𝑟 is the distance between the rear wear and the center of the gravity (COG). Reminding the angular speed is the variation of the angle respect the time 𝜔=𝜃󰇗(3.2) 32 Enhanced Sliding Mode Controls for Autonomous Vehicles 4.1.2. Second Control Law The second proposed sliding surface is 𝑠1=𝑥󰇗𝑒+𝑘1𝑥𝑒 (4.10𝑎) 𝑠2=𝑦󰇗𝑒+𝑘2𝑦𝑒+𝑘3sat(𝑦𝑒)𝜃𝑒(4.10𝑏) By the same procedure of preview section 4.1.1 the control signal (4.7) remains the same but the (4.8) is now 𝜃󰇗𝑒=−𝑞2−𝑝2sat(𝑠2)−𝑣󰇗sin(𝜃𝑒)+𝑥󰇗𝑒𝜔𝑟𝑒𝑓+𝑥𝑒𝜔󰇗𝑟𝑒𝑓−𝑘2𝑦󰇗𝑒 𝑣cos(𝜃𝑒)+𝑘3sat(𝑦𝑒)(4.12) The idea of this sliding surface is to enhance the steering angle which is more sensitive to the noise. 4.1.3. Third Control Law The third proposed sliding surface is 𝑠1=𝑥󰇗𝑒+𝑘1sgn(𝑥𝑒)𝑥𝑒 (4.13𝑎) 𝑠2=𝜃󰇗𝑒+𝑘2sgn(𝑦𝑒)𝑦𝑒+𝑘3sgn(𝜃𝑒)𝜃𝑒(4.13𝑏) The expression of (4.7) and (4.8) convert to 𝑣󰇗=−𝑞1−𝑝1sat(𝑠1)+𝑣sin(𝜃𝑒)𝜃󰇗𝑒−𝑦󰇗𝑒𝜔𝑟𝑒𝑓−𝑦𝑒𝜔󰇗𝑟𝑒𝑓+𝑣󰇗𝑟𝑒𝑓−𝑘1sgn(𝑥𝑒)𝑥󰇗𝑒 cos(𝜃𝑒) (4.14𝑎) 𝜃󰇗𝑒=−𝑞2−𝑝2sat(𝑠2)−𝑣󰇗sin(𝜃𝑒)+𝑥󰇗𝑒𝜔𝑟𝑒𝑓+𝑥𝑒𝜔󰇗𝑟𝑒𝑓−𝑘2sgn(𝑦𝑒)𝑦󰇗𝑒 𝑣cos(𝜃𝑒)+𝑘3sgn(𝜃𝑒) (4.14𝑏) The core of this controller design is to study how the sign of the parameter ki affect the control performance. Notice the saturation sat() function is applied instead of sgn() to reduce chattering. Chapter 4. Control Strategies 33 4.2. Linear Parameter Varying Model Predictive Control The LPVMPC method is designed using the dynamic model of the Section 3.2. The kinematic model is not used because if that the resulting control signal will be the linear and angular speed, and not the desired controls signal, the steering angle and acceleration. Converting the dynamic model from (3.18) in matrix form 𝑣󰇗𝑥=[−12𝐶𝑑𝜌𝐴𝑟𝑣𝑥2+𝜇𝑚𝑔 𝑚𝑣𝑥𝐶𝑓sin(𝛿) 𝑚𝑣𝑥𝑙𝑓𝐶𝑓sin(𝛿) 𝑚𝑣𝑥+𝑣𝑦][𝑣𝑥 𝑣𝑦 𝜔]+[−𝐶𝑓sin(δ) 𝑚1][𝛿𝑎](4.15𝑎) 𝑣󰇗𝑦=[0 −𝐶𝑓cos(𝛿)+𝐶𝑟 𝑚𝑣𝑥−𝐶𝑓𝑙𝑓cos(𝛿)−𝐶𝑟𝑙𝑟 𝑚𝑣𝑥−𝑣𝑥][𝑣𝑥 𝑣𝑦 𝜔]+[𝐶𝑓cos(δ) 𝑚0][𝛿𝑎] (4.15𝑐) 𝜔󰇗=[0𝐶𝑓sin(𝛿) 𝑚𝑣𝑥𝐶𝑓𝑙𝑓2cos(𝛿)+𝐶𝑟𝑙𝑟2 𝐼𝑣𝑥][𝑣𝑥 𝑣𝑦 𝜔]+[𝐶𝑓𝑙𝑓cos(δ) 𝐼0][𝛿𝑎] (4.15𝑐) The polytopic LPV model in discretized using (2.3) with the sampling time Ts 𝑥(𝑘+1)=∑𝜇𝑖(ϑ)(𝐴𝑖𝑥(𝑘)+𝐵𝑖𝑢(𝑘)) 𝑁 𝑖=1 (4.16𝑎) Where the scheduling variable ϑ is [δ, 𝑣𝑥, 𝑣𝑦] with length of 3 then N = 2^3=8, and 𝐴(ϑ)=[1+𝐴11𝑇𝑠𝐴12𝑇𝑠𝐴13𝑇𝑠 0 1+𝐴22𝑇𝑠𝐴23𝑇𝑠 0 𝐴32 1+𝐴33𝑇𝑠](4.16𝑏) 𝐴11=−12𝐶𝑑𝐴𝑟𝑣𝑥2+𝜇𝑚𝑔 𝑚𝑣𝑥 𝐴12=𝐶𝑓sin(𝛿) 𝑚𝑣𝑥 𝐴13=𝐶𝑓𝑙𝑓sin(𝛿) 𝑚𝑣𝑥+𝑣𝑦 (4.16𝑐) 𝐴22=−𝐶𝑓cos(𝛿)+𝐶𝑟 𝑚𝑣𝑥 𝐴23=−𝐶𝑓𝑙𝑓cos(δ)−𝐶𝑟𝑙𝑟 𝑚𝑣𝑥−𝑣𝑥(4.16𝑑) 𝐴32=−𝐶𝑓𝑙𝑓cos(𝛿)−𝐶𝑟𝑙𝑟 𝐼𝑣𝑥 𝐴33=−𝐶𝑓𝑙𝑓2cos(𝛿)+𝐶𝑟𝑙𝑟2 𝐼𝑣𝑥(4.16𝑒) And the matrix of control signal is 𝐵(ϑ)=[𝐵11 1 𝐵21 0 𝐵31 0]𝑇𝑠(4.16𝑓) 𝐵11=−𝐶𝑓sin(𝛿) 𝑚 𝐵21=𝐶𝑓cos(𝛿) 𝑚 𝐵31=𝐶𝑓𝑙𝑓cos(𝛿) 𝐼(4.16𝑔) 34 Enhanced Sliding Mode Controls for Autonomous Vehicles The discrete LPVMPC formulation is minimize 𝐽𝑘=∑(𝑥𝑘+𝑖 𝑇𝑄𝑥𝑘+𝑖+Δ𝑢𝑘+𝑖𝑅Δ𝑢𝑘+1) 𝐻𝑝−1 𝑖=1 +𝑥𝑘+𝑁 𝑇𝑄𝑥𝑘+𝐻𝑝 subject to: 𝑥𝑘+𝑖+1=∑𝐴𝑗𝑥𝑘+𝑖+𝐵𝑗𝑢𝑘+𝑖 𝐻𝑝 𝑗=1 𝑢𝑘+𝑖=𝑢𝑘+𝑖−1+Δ𝑢𝑘+𝑖 𝑖=0,1,…,𝐻𝑝−1 𝑢𝑚𝑖𝑛≤𝑢𝑘+𝑖≤𝑢𝑚𝑎𝑥 𝑖=0,1,…,𝐻𝑝−1 𝑥𝑚𝑖𝑛≤𝑥_{𝑘+𝑖}≤𝑥𝑚𝑎𝑥 𝑖=1,2,…,𝐻𝑝 (4.17) Where the increment of the control signal is optimized instead of directly the control signal. Chapter 5. Implementation 35 5. Implementation 5.1. Simulation Environment 5.1.1. Software The implementation of the work is done with Matlab 2020b. YALMIP Toolbox is used for the formulation and resolution of optimization problem, SeDuMi 1.3 is chosen as the numerical which is appropriated to solve convex problem. In the sense neither the lower level solver ‘quadprog’ and higher level solver ‘GeRuBi are not applicable for our optimization problem due to its characters; both solvers don’t return correct results if is forced to apply them. 5.1.2. Scenario The chosen scenario for test is shown in the Figure 5.1, which is a racing road that contains comprehensive road conditions, such like 90 and 180 degrees curves, straight line, and consecutive curves. Figure 5.1. Roadmap ‘Verschueren 2016’ from [9] with and without color. The marked parallel line on the upper side of the roadmap is the starting line, the vehicle will drive with clockwise sense and stop in the same starting line completing one laps. The simulation horizon has length of 446 steps with resolution of Ts = 0.02 seconds., which mean 8.92 second in the real world. But notice that higher resolution (0.001 seconds) is used when solving the ODE45 in order to make a correct vehicle simulation. Border line and outer region (green area of the Figure 5.1) are not obstacle which mean the vehicle can go outside of the circuit in the case bad control is applied. 36 Enhanced Sliding Mode Controls for Autonomous Vehicles 5.1.3. Simulation Model The simulation model is computed nonlinearly with the ODE45 function of the Matlab. Which return the integral value of (5.1). Figure 5.2. Vehicle model: Relevant states and variables. The model used for the simulation is the dynamic model with two increased states 𝑦𝑒 and 𝜃𝑒, corresponding to the error respect to the center line and the orientation error. 𝑑𝑣𝑥 𝑑𝑡 =𝑎−𝐶𝑓sin(𝛿) 𝑚𝛿+𝐶𝑓sin(𝛿) 𝑚𝑣𝑥𝑣𝑦+(𝑙𝑓sin(𝛿) 𝑚𝑣𝑥+𝑣𝑦)𝜔−12𝐶𝑑𝜌𝐴𝑟𝑣𝑥2 𝑚−𝜇𝑔 (5.1𝑎) 𝑑𝑣𝑦 𝑑𝑡=𝐶𝑓cos(𝛿) 𝑚𝛿−𝐶𝑓cos(𝛿)+𝐶𝑟 𝑚𝑣𝑥𝑣𝑦−(𝑙𝑟𝐶𝑓cos(𝛿)−𝑙𝑟𝐶𝑟 𝑚𝑣𝑥+𝑣𝑥)𝜔 (5.1𝑏) 𝑑𝜔 𝑑𝑡=𝐶𝑓𝑙𝑓cos(𝛿) 𝐼𝛿−𝐶𝑓𝑙𝑓cos(𝛿)−𝐶𝑟𝑙𝑟 𝐼𝑣𝑥𝑣𝑦−𝐶𝑓𝑙𝑓2cos(𝛿)+𝐶𝑟𝑙𝑟2 𝐼𝑣𝑥𝜔 (5.1𝑐) 𝑑𝑦𝑒 𝑑𝑡 =𝑣𝑥sin(𝜃𝑒)+𝑣𝑦cos(𝜃𝑒) (5.1𝑑) 𝑑𝜃𝑒 𝑑𝑡 =𝜔+𝜅𝑣𝑥sin(𝜃𝑒)−𝑣𝑦cos(𝜃𝑒) 1−𝑒𝑦𝜅 (5.1𝑒) The new element 𝜅 is the curvature of the road, which is a given data that change dependent the location of the vehicle in the map. The ODE45 return the integrated value of the model (5.1), which are [𝑣𝑥, 𝑣𝑦, 𝜔, 𝑦𝑒, 𝜃𝑒], the remains states are obtained by error based position model shown in (5.2). 𝑥=𝑥𝑟𝑒𝑓−𝑦𝑒sin(θ) (5.2𝑎) 𝑦=𝑦𝑟𝑒𝑓+𝑦𝑒sin(θ) (5.2𝑏) 𝜃=wrap(𝜃𝑟𝑒𝑓−𝜃𝑒) (5.2𝑐) Chapter 5. Implementation 37 Where the wrap() function ensures the angle not adopt incorrect value since its rotative value from 0 radians to 2π radians (0 degrees to 360 degrees). wrap(𝜃)={𝜃+2𝜋 𝜃<−𝜋 𝜃 −𝜋≤𝜃≤𝜋 𝜃−2𝜋 𝜃>𝜋 (5.3) The input control signal for the simulation model is the steering angle and the acceleration. 5.1.4. Planner The planner is the reference value to be tracked, depending on the control method, the required reference variable is different. Table 5.1. Planner availability and uses. Planner Availability Used in NLSMC Used in LPVMPC 𝑥𝑟𝑒𝑓 Yes Yes Yes 𝑦𝑟𝑒𝑓 Yes Yes Yes 𝜃𝑟𝑒𝑓 Yes Yes Yes 𝑣𝑥,𝑟𝑒𝑓 Yes Yes Yes 𝑣𝑦,𝑟𝑒𝑓 Yes Yes Yes 𝜔𝑟𝑒𝑓 Yes Yes Yes 𝛿𝑟𝑒𝑓 Yes No No 𝑎𝑟𝑒𝑓 Yes Yes No 𝜔󰇗𝑟𝑒𝑓 Yes Yes No 38 Enhanced Sliding Mode Controls for Autonomous Vehicles 5.1.5. Parametrization The physical nominal parameters of vehicle are shown in the Table 5.2, which corresponds to a small vehicle appropriate for the roadmap scenario size of Figure 5.1. Table 5.2. Physical parameter of the vehicle. Parameter Values [Units] Parameter Values [Units] 𝑙𝑟 0.125 [m] 𝑙𝑓 0.125 [m] 𝐶𝑟 65 [N/rad] 𝐶𝑓 65 [N/rad] 𝜇 0.05 𝐼 0.03 [Kg/m²] 𝑚 1.98 [Kg] 𝜌 1.225 [Kg/m³] 𝐴𝑟 1.91 [m²] 𝐶𝑑 0.36 Table 5.3. Dynamic state and control bound Variable Lower bound Upper bound Units 𝑣𝑥 0.1 20 [m/s] 𝑣𝑦 -1 1 [m/s] 𝜔 -1.42 1.42 [rad/s] δ -0.4 0.1 [rad] 𝑎 -1 2.5 [m/s²] The sampling time of LPVMPC is Ts = 0.02 seconds. Chapter 5. Implementation 39 5.2. Tuning Strategy 5.2.1. Fitness Function The tuning consists of the selection of the parameters value to obtain the best result of the fitness function. The most important of the tuning is the feasibility in the term of computational capacity of the computer and in the term of the time consumption. The chosen fitness function is the 2-norm of the root square mean error (RMSE) of the 𝑦𝑒 and 𝑣𝑥,𝑒 along the trajectory. min√𝑦𝑟𝑚𝑠𝑒 2+𝑣𝑥,𝑟𝑚𝑠𝑒 2(5.1) The kinematic (position) and dynamic (speed) performances in each x and y axis are evaluated while the rotation is implicitly considered in both variables. 5.2.2. Manual Tuning The manual consists of to adjust the parameters value increasing or decreasing regarding if the result is improved or not. There are no general criteria to guide how the parameters should be, but the Bryson’s rule [25] is a good initial value for start, that giving a reasonable simulation result. 𝑄𝑖=1 maximum acceptance value of (𝑥𝑖2) 𝑖=1,…,𝑛_𝑥 (5.2𝑎) 𝑅𝑗=1 maximum acceptance value of (𝑢𝑗2) 𝑗=1,…,𝑛_𝑢 (5.2𝑏) For the LPVMPC, following the value of Table 5.3, he initial values for Q and R are 𝑄= [ 1 20 0 0 0 1 0 0 0 1 1.42 ] 𝑅=[1 0.4 0 01 2.5](5.3) For the NLSMC the Bryson’s cannot easily applied because parameter such like q1 or q2 are applied to the sliding variable s1 and s2 which assumption for their bound is more complicated. The manual tuning is only implemented for LPVSMC and just for one Hp to illustrate. 40 Enhanced Sliding Mode Controls for Autonomous Vehicles 5.2.3. Pseudo-Greedy Search The greedy search is to test all the possible value and with this manner the global optimal is guaranteed but the cost of time could reach to infinite, like in the case of tuning, the selection of the value is inside the wide range {-∞ < x < ≤ ∞}, at same time the resolution of the value (the numbers decimals) is also theoretically infinite. Hence, to perform a feasible tuning, either the range or the resolution are bounded. Then the value of the parameter will be chosen from the candidate [0.01, 0.1, 1, 10, 100], in total 5^7 = 78125 simulation is performed for the each NLSMC, and, in total 5^5 = 3125 test is performed for each LPVMPC. 5.2.4. Genetic Algorithm In the genetic algorithm [26] the initial value is generated randomly with a specific number of sample (population), a percentile of those sample that return the best result will remain to the next iteration (generation) while remain sample are removed. In the next iteration, the population is refilled by crossover and mutation based on those remained elite sample of the preview iteration. In the crossover, two sample (parents) give part of their value, with a specific scale, and create a new sample merging their values. The mutation is when the number of variables is higher than 1, which implies each sample is a vector instead of a number, random sample will be modified due to the mutation, the modification could be in the sequence order or in the value. Figure 5.3. GA detailed flow chart. Chapter 5. Implementation 41 The advantage of GA is it can evolve as many generations as indicated, then the better result can be obtained, the drawback is relative high time consumption. By testing and observing the tuning results, the most relevant factor in the tunning is the random initial values should be good generated that can archive the convergence in later generations, then the tuning strategy is to run high quantity of tuning simulation and to find out statistically a good result. The main hyperparameter of the GA in Matlab are. - MaxGenerations: Maximum number of iterations before the algorithm halts. The default number is 100 time input number (number parameters in our case). - MaxTime: The algorithm stops after running for MaxTime seconds. Default value is inifite. - MaxStallGenerations: The algorithm stops if the average relative change in the best fitness function value over MaxStallGenerations generations is less than or equal to FunctionTolerance. Default setting is 50. - MaxStallTime: The algorithm stops if there is no improvement in the objective function for MaxStallTime seconds. Default value is infinite. - PopulationSize: Size of the population. The default number is 50 if the number of inputs variables are less or equal to 5, otherwise the default number is calculated as {min(max(10*nvars,40),100)}. The GA tuning for LPVMPC is not applicable due to its consumption cost. 5.2.5. Particle Swarm Optimization The particle swarm optimization the current value or position (P) of each sample by the weighted tendence speed (V) between the current local optimal value and the supposed global optimal value. 𝑃𝑖𝑡+1=𝑃𝑖𝑡+𝑉𝑖𝑡+1 (5.3𝑎) 𝑉𝑖𝑡+1=𝜔𝑉𝑖𝑡+𝑐1𝑟1(𝑃𝑙𝑜𝑐𝑎𝑙𝑏𝑒𝑠𝑡 𝑡−𝑃𝑖𝑡)+𝑐2𝑟2(𝑃𝑔𝑙𝑜𝑏𝑎𝑙𝑏𝑒𝑠𝑡 𝑡−𝑃𝑖𝑡) (5.3𝑏) Where the c1 and c2 are the corresponding self-weight and social weight. The r1 and r2 are random value uniformly selected from range {0, 1}. The ω is the inertia. With higher self-weight the sample tend to remain their position, with higher social weight the sample tend to concentrate to the same point [27]. 48 Enhanced Sliding Mode Controls for Autonomous Vehicles passing a curve and change radically. Even this controller reaches less tracking error. Finally, the third designed sliding surface has higher smoothness and only shown little chattering in few parts of the trajectory, which showing best control stability but with less precision in reference tracking than SF02. (a) Roadmap plot of SF01-PSO (b) Control signal of SF01-PSO (b) Roadmap plot of SF02-PSO (d) Control signal of SF02-PSO (e) Roadmap plot of SF03-PSO (f) Control signal of SF03-PSO Green square: starting position. Dot line: reference path. Color line: real path. Figure 6.1. Simulation plot of SF01-PSO, SF02-PSO and SF03-PSO. To avoid the remaining part chattering that SF03-PSO present, more depth design should be performed in sliding surface or additional filter could be applied. Chapter 6. Simulation 49 6.3. LPVMPC Table 6.3. Detailed result of LPVMPC Controllers Value [ Q1, Q2, Q3, R1, R2 ] ye [m] vx [m/s] HP01-pGS [ 1.0000, 0.0100, 100.0000, 100.0000, 0.0100] .2913 .0268 HP01-GA [611.9745, 441.3051, 683.9590, 284.3961, 5.7264] .0602 .0255 HP01-PSO [638.2115, 405.2853, 598.3163, 460.8507, 13.7731] .0165 .0498 HP03-MT [ 0.1150, 0.1163, 0.1408, 2.7500, 0.2000] .1081 .2302 HP03-pGS [100.0000, 0.0100, 0.0100, 0.1000, 0.0100] .0803 .0104 HP03-GA [938.8634, 222.6836, 129.1532, 934.2530, 458.2701] .0581 .0233 HP03-PSO [320.7068, 19.1579, 52.0740, 540.8807, 0.3339] .0661 .0112 HP09-pGS [ 1.0000, 0.0100, 0.0100, 0.1000, 100.0000] .3090 .0960 HP09-GA [ 51.1532, 0.3621, 1.2749, 489.7435, 0.6504] .4223 .0162 HP09-PSO [ 1.9765, 374.7504, 354.4971, 800.5463, 149.2229] .3751 .4600 In the Figure 6.2 illustrate the result of the LPVMPC tuned by with Bryson’s law, the curve of the control signal is very smooth specially the acceleration, because the control increment is constrained, but showing bad performance in the reference tracking, where critical error occur during the consecutive 180 degrees curves at right side of the roadmap, which mean the controller HP03-MT is not enough good to react to sophisticated road situation. (a) Roadmap plot of HP03-MT (b) Control signal of HP03-MT Green square: starting position. Dot line: reference path. Color line: real path. Figure 6.2. Simulation plot of HP03-MT. Nevertheless, taking result of LPVMPC tuned with GA shown in the Figure 6.3 (all plots are Annex B). For low Hp such the controller HP01-GA, the vehicle can follow the reference but showing relatively high frequency change in the control signal, because the controller is optimizing only for short horizon, and affecting the optimal result along the whole trajectory. 50 Enhanced Sliding Mode Controls for Autonomous Vehicles The HP03-GA show less changes as the Hp becomes higher and so balance the optimality between the prediction horizon and the whole simulation horizon. The controller HP09-GA show worse performance, this does not mean the LPVMPC is failed for having a large Hp, but the complexity of the optimization problem increases as the size of the Hp and so the difficulty to find out the correct tuned parameters. For current quantity of tuning simulation done is not enough for this LPVMPC. (a) Roadmap plot of HP01-GA (b) Control signal of HP01-GA (c) Roadmap plot of HP03-GA (d) Control signal of HP03-GA (e) Roadmap plot of HP09-GA (f) Control signal of HP09-GA Green square: starting position. Dot line: reference path. Color line: real path. Figure 6.3. Simulation plot of HP01-GA, HP03-GA, and HP09-GA. In addition, the HP01-PSO which shown lowest tracking error also has the same problem of high frequenche change in accletaion control (plot can be find in Annex B). Chapter 6. Simulation 51 6.4. Further simulation Longer simulation than one lap is preformed to compare their accumulative deviation in the non-tuned stage. The chosen controller is SF03-PSO and HP03-GA. (a) Roadmap plot of SF03-PSO (b) Control signal of SF03-PSO (c) Roadmap plot of HP03-GA (d) Control signal of HP03-GA Figure 6.4. Simulation until 600 steps. The NLSMC shown high quality in the reference tracking with the Best Cost increase only up to 0.1279 but chattering become serious problems, which is high in both frequency and magnitude. LPVMPC still show it smoothness in the control signal and the tracking performance is evidently degenerated in the control, consequently the Best Cost reached to 0.2992. 52 Enhanced Sliding Mode Controls for Autonomous Vehicles (a) Roadmap plot of SF03-PSO (b) Control signal of SF03-PSO (c) Roadmap plot of HP03-GA (d) Control signal of HP03-GA Figure 6.5. Simulation until 775 steps. If continue running, as shown in the Figure 6.5, in the final half lap, the NLSMC increased error more than LPVMPC, which passed from the Best Cost of 0.1279 to 0.1687. And the LPVMPC has a small increasement in the Best Cost from 02992 to 0.3235. Chapter 6. Simulation 53 6.5. Comparison The overview comparison of two controller is shown in below. Tracking. Both NLSMC and LPVMPC can reach very similar result in the tracking error. With HP01-PSO, the best result of LPVMPC, has 0.002 of advantage respect to SF02-PSO. Tuning. The NLSMC has 7 parameters, which implied more difficulty to be well tuned, but it is compensate by the efficiency in computation. For the LPVMPC, at contrary, only has 5 parameters but the time consuming control strategy derivate difficulty to the tuning. Applicability. The NLSMC has limitation to considering more state and behaviors of the real system, while MPCLPV can easily adopt it by expanding new state in the problem formulation without need to change the structure. For example, the error state 𝑦𝑒 and θ𝑒 of the simulation model (5.1) can be added to the LPVMPC, by the cost of size of the parameter matrix Q is also expanded correspondingly and therefore the cost to tuning. Dependency. The NLSMC require two planner variables more than the LPVMPC which are the acceleration and the angular acceleration, where the former is one of the control signals, this restricts the applicability of NLSMC and more requirements to the planner for the NLSMC implementation. Bounding. The NLSMC is constraint free control method, therefore the physical and performance constraints cannot be considered such like increment rate of the control signal to avoid definitely the chattering, at opposite, the LPVMPC are able to consider many constraints as specified but the formulation of complex problem could reach to infeasibility for the solver. 54 Enhanced Sliding Mode Controls for Autonomous Vehicles 7. Planning This TFM Project amount 12 ECTS (European Credit Transfer and Accumulation System), which for the Master of Automatic and Robotic, each ECTS is equivalent to 30 working hours, in total 360 hours. Considering part time job, 20 hour is performed per week. In total the project schedule has horizon of 3 months, i.e., 12 working weeks. Table 7.1. Time Schedule. Month 1 Month 2 Month 3 01 02 03 04 05 06 07 08 09 10 11 12 Research NLSMC Design Tuning LPVMPC Design Tuning Documentation Chapter 8. Project Budget 55 8. Project Budget In this section, the project budget of this TFM is accounted with the criteria to give a reference of time and money for another individual, company or research team if they want are considering reproducing this work. Table 8.1. Economical budget for the development of the project. Items Costs [€/hour] Time [hours] Amount [€] Matlab: standard license 0.1027*1 360 hours*2 36.99 YALMIP license - - 0.00 Engineer wage 15.38 5608.80 Azure: cloud service (D3) x2 2x 0.5179 253 hours*3 131.03 Laptop amortization 0.0262*4 360 hours 9.45 Total 5786.27 1* Matlab has only annual or perceptual options, in this case the annual is considered, which implies 900 euro per year (365 days). 2* The TFM Project of 12 ECTS is equaling to 360 working hours. 3* The amount time of Table 6.1 is 244 hours, but more time is used for preview testing. 4* Laptop (Hp envy 13) has cost of 1150,00 euro, the amortization is linearly distribution in 5 years. The total cost of work amounts to 5786.27 euro, which if we add the corresponding VAT (21%) would be 7001.39 euro. 56 Enhanced Sliding Mode Controls for Autonomous Vehicles 9. Economy, environment, society, and equality 9.1. Economy Increase traffic efficiency and safety. Numerous studies concluded that if a significant rate of AV were introduced in the traffic, it would reduce the congestion and increase efficiency for having a planned and smooth road guidance and driving. But it also depends on the capacity of the autonomous vehicle, if it’s a low capacity passenger like personalized AV only with capacity of one person, then the congestion will increase consequently [28] [29]. The statistical model based studies [30] of road accidents show that human error is the main cause. The use of AV can avoid several traffic accidents causes by the lack of attention; of human driver, but the complete evaluation to level of security of AV faces technical challenges, and this is one of the reasons that AV still are not totally able to drive in highway. On the other hand, the high voltage battery installed in the AV causes new different efficiency and security problems. The charging convenience depending on the local infrastructure, an EV parking cannot be so compact as normal parking due to the flame isolation requirements, and fireproof car cover are needed, which all implied increase of cost for the government, charging companies and consumers. Reduction of driver employer number. In [3] examines people’s views on the AV employment impacts through an online survey of 773 respondents from 50 countries. The survey found that age, field of work/study, self-perceived level of AV awareness, income, and gender influence respondents’ views on the transportation labor market in the AV era. Low-skilled employees, non-transportation professionals, young people, and those who are unaware of AV are more likely to believe that AV may lead to large-scale layoffs in the industry [31]. Conversely, the job opportunity of AV field and of the professional driver like race driver will increase accordingly the percentage of the AV and driving license holding rate. 9.2. Environment Carbon emissions. The contamination impact of AV is evaluated mainly by the energy consumption and emissions levels. An AV normally use electric or hybrid power, which show significantly reduce carbon emissions respect the traditional combustible vehicle [32]. Furthermore, its low noise is an important consideration in the urban environment, under conditions fulfilling the minimum pass-by noises requirements. However, the AV and EV Chapter 9. Economy, environment, society, and equality 57 may cause light pollution. Battery Recycling. As all EV, most important treatments for the AV are the battery. The lithium batteries degradation poses environmental and human health risks and affects the sustainable development of society. The [33] provides several recommendations like development of new recycling technologies, increasing the value of recycled cathode materials, adapting battery systems for recycling, and building intelligent automated robots replacement. 9.3. Society Transport habit. Most of the current implementation of the AV is found in the public transport, the implementation of the AV will imply the increase of use of public transport such like small size bus and passenger car. For personal AV, the owner during the driving will need not to pay hundreds of percent of attention in the control and will have more time to do other tasks, this time saving will promote people to use more AV [34]. Legislative. Currently the AV is not allowed to drive freely in the road and highway without monitoring. Thereby, there is still no homologation for AV. In the technical point of view the homologation test is can be easily implemented, furthermore the characteristic of AV, like stable speed control or driverless, can help to perform the test in easier and less laborious manner. However, the main difficult for the homologation is found in the legal field, the critical issue is how to determine the responsibility party in a traffic accident of AV [35]. 9.4. Equality Genger. One of the gender stereotypes is that “women conduct worse than man”, and the social stereotype will affect the induvial in the society to adapt the stereotype as the [36] shown. It is difficult to eliminate the existing effect of the stereotype, but the use of AV where the driverless aspect can take human out of the gender context in the driving [34]. Disability. Depending on the country, person with low level disability is allowed for having driving license, but not for all, the implementation of AV can eliminate discrimination of the capacity of driving in the society and to care and help people with disability [34]. As same manner by taking out of context in the capacity of driving. The same criteria can be applied to the driving age limitation for the senior or underage people, driver professionality discrimination in salary and helping to eliminate the regional driving license issue, e.g., a person who has US driving license is not allowed to drive in EU. 64 REFERENCES [30] T. Winkle, Safety Benefits of Automated Vehicles: Extended Findings from Accident Research for Development, Validation and Testing, 2016. [31] A. Nikitas, A.-E. Vitel and C. Cotet, Autonomous vehicles and employment: An urban futures revolution or catastrophe?, 2021. [32] Ó. Silva, R. Cordera, E. González-González and S. Nogués, Environmental impacts of autonomous vehicles: A review of the scientific literature, 2022. [33] X. Yu, W. Li, V. Gupta, H. Gao, D. Tran, S. Sarwar and Z. Chen, Current Challenges in Efficient Lithium-Ion Batteries’ Recycling: A Perspective, 2022. [34] D. Bissell, T. Birtchnell, E. L. Hsu and A. Elliott, Autonomous automobilities: The social impacts of driverless vehicles, 2018. [35] I. Begishev, D. Bersei, L. Sherbakova, R. Zhirov and O. Kolesnikova, Problems of legal regulation of unmanned vehicles, 2022. [36] A. Moè, M. Cadinu and A. Maass, Women drive better if not stereotyped, 2015. [37] J. Liu, Sliding Mode Control Using MATLAB, ELSEVIER, 2017 . [38] M. R. Rajanna, Sliding Mode Control for Autonomous Vehicle, 2017. ANNEX 65 Annex Annex A. Complete plot of NLSMC SF01-pGS SF02-pGS SF03-pGS 66 ANNEX SF01-GA SF02-GA SF03-GA ANNEX 67 SF01-PSO SF02-PSO SF03-PSO 68 ANNEX Annex B. Complete plot of LPVSMC HP01-pGS HP03-pGS HP09-pGS ANNEX 69 HP01-GA HP03-GA HP09-GA 70 ANNEX HP01-PSO HP03-PSO HP09-PSO