DOCSLIB.ORG

Decentralized Optimal Control of Connected and Automated Vehicles in a Corridor

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Decentralized Optimal Control of Connected and Automated Vehicles in a Corridor 2018 21st International Conference on Intelligent Transportation Systems (ITSC) Maui, Hawaii, USA, November 4-7, 2018 Decentralized Optimal Control of Connected and Automated Vehicles in a Corridor Liuhui Zhao Member, IEEE, Andreas A. Malikopoulos, Senior Member, IEEE Abstract— In earlier work, we established a decentralized op- traffic operation center and received control actions from the timal control framework for coordinating online connected and center to improve network-wide traffic flow [11], [12]. automated vehicles (CAVs) in specific transportation segments, In earlier work, a decentralized optimal control framework e.g., urban intersections, merging roadways, roundabouts, and speed reduction zones. In this paper, we address coordination was established for coordinating online CAVs in different of CAVs in a corridor with multiple such scenarios and derive a transportation segments. A closed-form, analytical solution closed-form analytical solution that includes interior boundary without considering state and control constraints was pre- conditions. We evaluate the effectiveness of the solution through sented in [13], [14], and [15] for coordinating online CAVs simulation in VISSIM. The proposed approach reduces signif- at highway on-ramps, in [16] at two adjacent intersections, icantly both fuel consumption and travel time for the CAVs compared to the baseline scenario where traditional human- and in [17] at roundabouts. The solution of the unconstrained driven vehicles without control are considered. problem was also validated experimentally at the University of Delaware’s Scaled Smart City using ten robotic CAVs [18] in a merging roadway scenario. The solution of the optimal I. INTRODUCTION control problem considering state and control constraints was Urban intersections, merging roadways, roundabouts, and presented in [19] at an urban intersection without considering speed reduction zones along with the driver responses to rear-end collision avoidance constraint, and the conditions various disturbances [1] are the primary sources of bottle- under which the latter does not become active were presented necks that contribute to traffic congestion. Connectivity and in [20]. automation in vehicles provide the most intriguing oppor- In this paper, we consider coordination of a number of tunity for enabling users to better monitor transportation CAVs through a corridor. To ensure that no lateral collision network conditions and make better operating decisions. occurs, we impose interior constraints in our Hamiltonian Several research efforts have been reported in the literature analysis and derive the optimal solution throughout the entire proposing different approaches on coordinating CAVs at corridor. different transportation segments, e.g., urban intersections, The paper is organized as follows. In Section II, we merging roadways, roundabouts, and speed reduction zones, formulate the problem and provide the modeling framework. with the intention to improve transportation efficiency. In In Section III, we derive the analytical, closed form solution 2004, Dresner and Stone [2] proposed the use of the reserva- with interior constraints. In Section IV, we validate the tion scheme to control a single intersection of two roads with effectiveness of the analytical solution in a simulation envi- vehicles traveling with similar speed on a single direction on ronment and conduct a comparison analysis with traditional each road, i.e., no turns are allowed. Since then, numerous human-driven vehicles. Finaly, the concluding remarks and approaches have been reported in the literature [3]–[5], to discussion are provided in Section V. achieve safe and efficient control of traffic through intersec- II. PROBLEM FORMULATION tions including extensions of the reservation scheme in [2]. Some approaches have focused on coordinating vehicles at We consider a corridor (Fig. 1) that consists of three intersections to improve the traffic flow [6]–[8]. A detailed merging zones, e.g., two merging roadways and one urban discussion of the research efforts in this area that have been intersection. The corridor has a coordinator that can monitor reported in the literature to date can be found in [9]. the vehicles traveling along the corridor within a control Although previous research aimed at enhancing our un- zone (shown with a dashed box in Fig. 1). Note that the derstanding of improving emerging transportation systems coordinator serves as an information center which is able to just a few efforts have reported results on corridors [10] that collect vehicular data through vehicle-to-infrastructure and is include multiple intersections and merging roadways. More not involved in any decision on the vehicle operation. Road recently, a control framework was developed for a coordi- side units could be placed in each merging zone and used nated and integrated corridor management in a mixed traffic to transmit data between vehicles and the coordinator. Thus, environment, where CAVs send information to a centralized the coverage of the coordinator is flexible and the length of corridor could be extended in the presence of connected This research was supported by ARPAE’s NEXTCAR program under the infrastructure. award number DE-AR0000796. Let N(t) 2 N be the number of CAVs in the corridor at The authors are with the Department of Mechanical Engineering, Uni- + versity of Delaware, Newark, DE 19716 USA (emails: [email protected]; time t 2 R and M 2 M be the number of merging zones [email protected].) along the corridor where lateral collisions may occur. When 978-1-7281-0322-8/18/$31.00 ©2018 IEEE 1252 To ensure that the control input and vehicle speed are within a given admissible range, the following constraints are imposed. ui;min ≤ ui(t) ≤ ui;max; and 0 f (2) 0 ≤ vmin ≤ vi(t) ≤ vmax; 8t 2 [ti ; ti ]; where ui;min, ui;max are the minimum deceleration and maximum acceleration for each vehicle i = 1;:::;N(t) 2 N, and vmin, vmax are the minimum and maximum speed limits respectively. To ensure the absence of rear-end collision of two con- secutive vehicles traveling on the same lane, the position of the preceding vehicle should be greater than or equal to Fig. 1. Corridor with connected and automated vehicles. the position of the following vehicle plus a predefined safe distance δi(t). Thus we impose the rear-end safety constraint a vehicle enters the control zone of the corridor, it broadcasts 0 f si(t) = ξi · (pk(t) − pi(t)) ≥ δi(t); 8t 2 [ti ; ti ]; (3) its origin-destination (OD) to the coordinator. Then, the coordinator assigns a unique number i = 1;:::;N(t) 2 N where si(t) 2 Si denotes the distance of vehicle i from the (Fig. 1) that serves as an identification of the vehicles within vehicle k which is physically located ahead of i. Relate the the control zone. The policy through which the sequence minimum safe distance δi(t) as a function of speed vi(t), that each vehicle crosses each merging zone throughout the 0 f corridor may be the result of a higher level optimization δi(t) = γi + ρi · vi(t); 8t 2 [ti ; ti ]; (4) problem, which is not addressed in this paper. In what where γ is the standstill distance, and ρ is minimum time follows, we will adopt a specific scheme for determining i i gap that vehicle i would maintain while following another this sequence of the vehicles crosses each merging zone vehicle. throughout the corridor but we emphasize that our analysis is not restricted to this sequence. B. Policy for Vehicle Sequence Crossing the Merging Zones To avoid any possible lateral collision while the vehicles In the modeling framework described above, we assume crossing the merging zones, once a vehicle i enters the that the vehicles traveling in the corridor do not change control zone, it computes the time that will be entering each lanes except to make necessary turns. When a vehicle i merging zone which is discussed in the next Section. Let t0 i enters the control zone of the corridor it computes the time be the initial time that vehicle i enters the control zone of the mj m t for each merging zone j based on the following three corridor, t j be the time that vehicle i enters the merging i i subsets: 1) S contains all vehicles share the same route zone j, j 2 M, and tf be the time that vehicle i exits the last i i with vehicle i, 2) L contains all vehicles that travel in the merging zone along its route. In what follows, we provide a i same lane in merging zone j with vehicle i but travel from policy for the vehicle sequence crossing the merging zones different routes (i.e., may have rear-end collision with vehicle in the corridor. The focus is on the lower level control i in merging zone j, but lateral collision in the immediate problem that yields for each vehicle the optimal control input upstream merging zone j −1), and 3) C contains all vehicles (acceleration/deceleration) to achieve the assigned tmj using i i from different entry links in merging zone j that may have a given policy that designates tmj (upon arrival of CAV i at i lateral collision with vehicle i. the entry of the control zone). mj To ensure that (3) is satisfied at ti , we first impose the A. Vehicle Model and Constraints following condition For simplicity, each vehicle is modeled as a double inte- m m p (t) p (t) p (t) t j = max min t j +ρ ; j +t0 ; j +t0; j +t0 ; grator i k i i 0 i i vmin vi vmax (5) p_i = vi(t) mj (1) where tk is the time when the vehicle k enters merging v_i = ui(t) zone j, k 2 Si, pj(t) is the distance from the entry point of 0 where pi(t) 2 Pi, vi(t) 2 Vi, and ui(t) 2 Ui denote the the control zone until the entry of merging zone j, and vi is position, speed and acceleration/deceleration (control input) the initial speed of vehicle i when it enters the control zone of each vehicle i in the corridor.
Load more

Recommended publications
  • Propt Product Sheet
    PROPT - The world’s fastest Optimal Control platform for MATLAB. PROPT - ONE OF A KIND, LIGHTNING FAST SOLUTIONS TO YOUR OPTIMAL CONTROL PROBLEMS! NOW WITH WELL OVER 100 TEST CASES! The PROPT software package is intended to When using PROPT, optimally coded analytical solve dynamic optimization problems. Such first and second order derivatives, including problems are usually described by: problem sparsity patterns are automatically generated, thereby making it the first MATLAB • A state-space model of package to be able to fully utilize a system. This can be NLP (and QP) solvers such as: either a set of ordinary KNITRO, CONOPT, SNOPT and differential equations CPLEX. (ODE) or differential PROPT currently uses Gauss or algebraic equations (PAE). Chebyshev-point collocation • Initial and/or final for solving optimal control conditions (sometimes problems. However, the code is also conditions at other written in a more general way, points). allowing for a DAE rather than an ODE formulation. Parameter • A cost functional, i.e. a estimation problems are also scalar value that depends possible to solve. on the state trajectories and the control function. PROPT has three main functions: • Sometimes, additional equations and variables • Computation of the that, for example, relate constant matrices used for the the initial and final differentiation and integration conditions to each other. of the polynomials used to approximate the solution to the trajectory optimization problem. The goal of PROPT is to make it possible to input such problem • Source transformation to turn descriptions as simply as user-supplied expressions into possible, without having to worry optimized MATLAB code for the about the mathematics of the cost function f and constraint actual solver.
    [Show full text]
  • Optimal Control Theory Version 0.2
    An Introduction to Mathematical Optimal Control Theory Version 0.2 By Lawrence C. Evans Department of Mathematics University of California, Berkeley Chapter 1: Introduction Chapter 2: Controllability, bang-bang principle Chapter 3: Linear time-optimal control Chapter 4: The Pontryagin Maximum Principle Chapter 5: Dynamic programming Chapter 6: Game theory Chapter 7: Introduction to stochastic control theory Appendix: Proofs of the Pontryagin Maximum Principle Exercises References 1 PREFACE These notes build upon a course I taught at the University of Maryland during the fall of 1983. My great thanks go to Martino Bardi, who took careful notes, saved them all these years and recently mailed them to me. Faye Yeager typed up his notes into a first draft of these lectures as they now appear. Scott Armstrong read over the notes and suggested many improvements: thanks, Scott. Stephen Moye of the American Math Society helped me a lot with AMSTeX versus LaTeX issues. My thanks also to Atilla Yilmaz for spotting lots of typos and errors, which I have corrected. I have radically modified much of the notation (to be consistent with my other writings), updated the references, added several new examples, and provided a proof of the Pontryagin Maximum Principle. As this is a course for undergraduates, I have dispensed in certain proofs with various measurability and continuity issues, and as compensation have added various critiques as to the lack of total rigor. This current version of the notes is not yet complete, but meets I think the usual high standards for material posted on the internet. Please email me at [email protected] with any corrections or comments.
    [Show full text]
  • Ensemble Kalman Filtering for Inverse Optimal Control Andrea Arnold and Hien Tran, Member, SIAM
    Ensemble Kalman Filtering for Inverse Optimal Control Andrea Arnold and Hien Tran, Member, SIAM Abstract—Solving the inverse optimal control problem for In the Bayesian framework, unknown parameters are mod- discrete-time nonlinear systems requires the construction of a eled as random variables with probability density functions stabilizing feedback control law based on a control Lyapunov representing distributions of possible values. The EnKF is a function (CLF). However, there are few systematic approaches available for defining appropriate CLFs. We propose an ap- nonlinear Bayesian filter which uses ensemble statistics in proach that employs Bayesian filtering methodology to param- combination with the classical Kalman filter equations for eterize a quadratic CLF. In particular, we use the ensemble state and parameter estimation [9]–[11]. The EnKF has been Kalman filter (EnKF) to estimate parameters used in defining employed in many settings, including weather prediction the CLF within the control loop of the inverse optimal control [12], [13] and mathematical biology [11]. To the authors’ problem formulation. Using the EnKF in this setting provides a natural link between uncertainty quantification and optimal knowledge, this is the first proposed use of the EnKF in design and control, as well as a novel and intuitive way to find inverse optimal control problems. The novelty of using the the one control out of an ensemble that stabilizes the system the EnKF in this setting allows us to generate an ensemble of fastest. Results are demonstrated on both a linear and nonlinear control laws, from which we can then select the control law test problem. that drives the system to zero the fastest.
    [Show full text]
  • Optimal Control Theory
    Optimal Control Theory Mrinal Kanti Ghosh Department of Mathematics Indian Institute of Science Bangalore 560 012 email : [email protected] 1 Introduction Optimal Control Theory deals with optimization problems involving a controlled dynamical system. A controlled dynamical system is a dynamical system in which the trajectory can be altered continuously in time by choosing a control parameter u(t) continuously in time. A (deterministic) controlled dynamical system is usually governed by an ordinary differential equation of the form 9 x_(t) = f(t; x(t); u(t)); t > 0 = (1:1) d ; x(0) = x0 2 R where u : Rd ! U is a function called the control, U a given set called the control set, and d d f : R+ × R × U ! IR : By choosing the value of u(t), the state trajectory x(t) can be controlled. The objective of controlling the state trajectory is to minimize a certain cost associated with (1.1). Consider the following important problem studied by Bushaw and Lasalle in the field of aerospace engineering. Let x(t) represent the deviation of a rocket trajectory from some desired flight path. Then, provided the deviation is small, it is governed to a good approxi- mation by 9 x_(t) = A(t)x(t) + B(t)u(t); t > 0 = x(0) = x0; x(T ) = 0; ; (1:2) u(t) 2 [−1; 1]; where A and B are appropriate matrix valued functions. The vector x0 is the initial deviation, u(t) is the rocket thrust at time t, and T is the final time.
    [Show full text]
  • PSO-Based Soft Lunar Landing with Hazard Avoidance: Analysis and Experimentation
    aerospace Article PSO-Based Soft Lunar Landing with Hazard Avoidance: Analysis and Experimentation Andrea D’Ambrosio 1,* , Andrea Carbone 1 , Dario Spiller 2 and Fabio Curti 1 1 School of Aerospace Engineering, Sapienza University of Rome, Via Salaria 851, 00138 Rome, Italy; [email protected] (A.C.); [email protected] (F.C.) 2 Italian Space Agency, Via del Politecnico snc, 00133 Rome, Italy; [email protected] * Correspondence: [email protected] Abstract: The problem of real-time optimal guidance is extremely important for successful au- tonomous missions. In this paper, the last phases of autonomous lunar landing trajectories are addressed. The proposed guidance is based on the Particle Swarm Optimization, and the differ- ential flatness approach, which is a subclass of the inverse dynamics technique. The trajectory is approximated by polynomials and the control policy is obtained in an analytical closed form solution, where boundary and dynamical constraints are a priori satisfied. Although this procedure leads to sub-optimal solutions, it results in beng fast and thus potentially suitable to be used for real-time purposes. Moreover, the presence of craters on the lunar terrain is considered; therefore, hazard detection and avoidance are also carried out. The proposed guidance is tested by Monte Carlo simulations to evaluate its performances and a robust procedure, made up of safe additional maneuvers, is introduced to counteract optimization failures and achieve soft landing. Finally, the whole procedure is tested through an experimental facility, consisting of a robotic manipulator, equipped with a camera, and a simulated lunar terrain. The results show the efficiency and reliability Citation: D’Ambrosio, A.; Carbone, of the proposed guidance and its possible use for real-time sub-optimal trajectory generation within A.; Spiller, D.; Curti, F.
    [Show full text]
  • ID 467 Optimal Control Applied to an Economic Model for Reducing Unemployment Problems in Bangladesh
    Proceedings of the International Conference on Industrial Engineering and Operations Management Bangkok, Thailand, March 5-7, 2019 Optimal Control Applied to an Economic Model for Reducing Unemployment Problems in Bangladesh Md. Haider Ali Biswas, Uzzwal Kumar Mallick, Mst. Shanta Khatun Mathematics Discipline, Khulna University Khulna-9208, Bangladesh [email protected],[email protected],[email protected] Kazi Masum Parves Joint Director, Bangladesh Bank, Khulna, Bangladesh [email protected] Abstract In this paper, a nonlinear mathematical model of unemployment is described and analyzed for reduction of unemployed persons using two controls ‘creation of new vacancies’ and ‘skilled manpower’. For maximization of the number of employed people at the end of implemented policy, we find the control profile of two controls using Pontryagin’s Maximum Principle. This study concludes two effective controls so that the number of unemployed persons at the end of implemented policies of the government is minimized while minimizing the systematic cost of policy making. Our numerical results as illustrated in the figures strongly agree with the analytical findings. Keywords: Unemployment, Socio-economic damage, Mathematical model, Optimal control, Numerical simulation. 1. Introduction Unemployment problems have become the most immense concerns all over the world. This issue is significantly an alarming concern in Bangladesh as well. Unemployment can be defined as a state of workless for a person, who is fit, capable and willing to work. It is well known that the main reason for socio-economic damage, deletion of morality and social values is unemployment. With the increasing tendency of unemployed population in a country significantly affects the other factors, such that the income per person, health costs, quality of healthcare and poverty and seriously hamper the growth of national economy.
    [Show full text]
  • Optimal Control the Optimal Control Problem Can Be Described By
    Optimal Control The optimal control problem can be described by introducing the system dynamics x_ = F (x; u) which is assumed to start in an initial state given by x(0) = x0 and has controllable parameters u u 2 U The objective function consists of a function of the final state [x(T )] and a cost function (or loss function) ` that is integrated over time. Z T J = [x(T )] + `(u; x)dt 0 1 The simplest optimization problem is depicted in Figure 1: min F~(x) x 40 4 20 2 0 0 -4 -2 -2 0 2 -4 4 Figure 1: A simple case for optimization: a function of two variables has a single minimum at the point x∗. 2 Constraints The simplest kind of constraint is an equality constraint, for example, g(x) = 0. Formally this addition is stated as min F~(x) subject to G(x) = 0 x This complication can be handled by the method of Lagrange. This method reduces the constrained problem to a new, unconstrained minimization prob- lem with additional variables. The additional variables are known as Lagrange multipliers. To handle this problem, add g(x) to f~(x) using a Lagrange mul- tiplier λ: F (x; λ) = F~(x) + λG(x) The Lagrange multiplier is an extra scalar variable, so the number of degrees of freedom of the problem has increased, but the advantage is that now sim- ple, unconstrained minimization techniques can be applied to the composite function. The problem becomes min F (x; λ) x,λ Most often the equality constraint is the dynamic equations of the system itself, x_ = F (x; u), and the problem is to find an optimal trajectory x(t).
    [Show full text]
  • Optimal and Autonomous Control Using Reinforcement Learning: a Survey
    2042 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 29, NO. 6, JUNE 2018 Optimal and Autonomous Control Using Reinforcement Learning: A Survey Bahare Kiumarsi , Member, IEEE , Kyriakos G. Vamvoudakis, Senior Member, IEEE , Hamidreza Modares , Member, IEEE , and Frank L. Lewis, Fellow, IEEE Abstract — This paper reviews the current state of the art on unsupervised, or reinforcement, depending on the amount and reinforcement learning (RL)-based feedback control solutions quality of feedback about the system or task. In supervised to optimal regulation and tracking of single and multiagent learning, the feedback information provided to learning algo- systems. Existing RL solutions to both optimal H2 and H∞ control problems, as well as graphical games, will be reviewed. rithms is a labeled training data set, and the objective is RL methods learn the solution to optimal control and game to build the system model representing the learned relation problems online and using measured data along the system between the input, output, and system parameters. In unsu- trajectories. We discuss Q-learning and the integral RL algorithm pervised learning, no feedback information is provided to the as core algorithms for discrete-time (DT) and continuous-time algorithm and the objective is to classify the sample sets to (CT) systems, respectively. Moreover, we discuss a new direction of off-policy RL for both CT and DT systems. Finally, we review different groups based on the similarity between the input sam- several applications. ples. Finally, reinforcement learning (RL) is a goal-oriented Index Terms — Autonomy, data-based optimization, learning tool wherein the agent or decision maker learns a reinforcement learning (RL).
    [Show full text]
  • Tomlab Product Sheet
    TOMLAB® - For fast and robust large- scale optimization in MATLAB® The TOMLAB Optimization Environment is a powerful optimization and modeling package for solving applied optimization problems in MATLAB. TOMLAB provides a wide range of features, tools and services for your solution process: • A uniform approach to solving optimization problem. • A modeling class, tomSym, for lightning fast source transformation. • Automatic gateway routines for format mapping to different solver types. • Over 100 different algorithms for linear, discrete, global and nonlinear optimization. • A large number of fully integrated Fortran and C solvers. • Full integration with the MAD toolbox for automatic differentiation. • 6 different methods for numerical differentiation. • Unique features, like costly global black-box optimization and semi-definite programming with bilinear inequalities. • Demo licenses with no solver limitations. • Call compatibility with MathWorks’ Optimization Toolbox. • Very extensive example problem sets, more than 700 models. • Advanced support by our team of developers in Sweden and the USA. • TOMLAB is available for all MATLAB R2007b+ users. • Continuous solver upgrades and customized implementations. • Embedded solvers and stand-alone applications. • Easy installation for all supported platforms, Windows (32/64-bit) and Linux/OS X 64-bit For more information, see http://tomopt.com or e-mail [email protected]. Modeling Environment: http://tomsym.com. Dedicated Optimal Control Page: http://tomdyn.com. Tomlab Optimization Inc. Tomlab
    [Show full text]
  • Learning, Expectations Formation, and the Pitfalls of Optimal Control Monetary Policy
    FEDERAL RESERVE BANK OF SAN FRANCISCO WORKING PAPER SERIES Learning, Expectations Formation, and the Pitfalls of Optimal Control Monetary Policy Athanasios Orphanides Central Bank of Cyprus John C. Williams Federal Reserve Bank of San Francisco April 2008 Working Paper 2008-05 http://www.frbsf.org/publications/economics/papers/2008/wp08-05bk.pdf The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Federal Reserve Bank of San Francisco or the Board of Governors of the Federal Reserve System. Learning, Expectations Formation, and the Pitfalls of Optimal Control Monetary Policy Athanasios Orphanides Central Bank of Cyprus and John C. Williams¤ Federal Reserve Bank of San Francisco April 2008 Abstract This paper examines the robustness characteristics of optimal control policies derived un- der the assumption of rational expectations to alternative models of expectations. We assume that agents have imperfect knowledge about the precise structure of the economy and form expectations using a forecasting model that they continuously update based on incoming data. We ¯nd that the optimal control policy derived under the assumption of rational expectations can perform poorly when expectations deviate modestly from rational expectations. We then show that the optimal control policy can be made more robust by deemphasizing the stabilization of real economic activity and interest rates relative to infla- tion in the central bank loss function. That is, robustness to learning provides an incentive to employ a \conservative" central banker. We then examine two types of simple monetary policy rules from the literature that have been found to be robust to model misspeci¯cation in other contexts.
    [Show full text]
  • Numerical Methods for Solving Optimal Control Problems
    University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 5-2015 Numerical Methods for Solving Optimal Control Problems Garrett Robert Rose University of Tennessee - Knoxville, [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Numerical Analysis and Computation Commons Recommended Citation Rose, Garrett Robert, "Numerical Methods for Solving Optimal Control Problems. " Master's Thesis, University of Tennessee, 2015. https://trace.tennessee.edu/utk_gradthes/3401 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Garrett Robert Rose entitled "Numerical Methods for Solving Optimal Control Problems." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Mathematics. Charles Collins, Major Professor We have read this thesis and recommend its acceptance: Abner Jonatan Salgado-Gonzalez, Suzanne Lenhart Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) Numerical Methods for Solving Optimal Control Problems A Thesis Presented for the Master of Science Degree The University of Tennessee, Knoxville Garrett Robert Rose May 2015 1 Dedication To my parents and brother “If you’re into that sort of thing.” ii Acknowledgements I would like to take a second to thank those who help with either this thesis, or my academic career in general.
    [Show full text]
  • Linear-Quadratic Optimal Control in Maximal Coordinates
    Linear-Quadratic Optimal Control in Maximal Coordinates Jan Brudigam¨ 1 and Zachary Manchester2 (a) Acrobot basins of attraction (b) Acrobot Abstract— The linear-quadratic regulator (LQR) is an effi- 3.14 cient control method for linear and linearized systems. Typi- cally, LQR is implemented in minimal coordinates (also called generalized or “joint” coordinates). However, other coordinates θ1 are possible and recent research suggests that there may be 1.57 numerical and control-theoretic advantages when using higher- u dimensional non-minimal state parameterizations for dynamical 0 systems. One such parameterization is maximal coordinates, in θ which each link in a multi-body system is parameterized by 2 its full six degrees of freedom and joints between links are -1.57 modeled with algebraic constraints. Such constraints can also max. represent closed kinematic loops or contact with the environ- (min.) ment. This paper investigates the difference between minimal- both and maximal-coordinate LQR control laws. A case study of -3.14 applying LQR to a simple pendulum and simulations comparing 0 1.57 3.14 4.71 6.28 the basins of attraction and tracking performance of minimal- and maximal-coordinate LQR controllers suggest that maximal- Fig. 1: Basins of attraction for maximal- and minimal-coordinate coordinate LQR achieves greater robustness and improved LQR when stabilizing the upright equilibrium (black cross) of tracking performance compared to minimal-coordinate LQR an acrobot from different initial configurations. The maximal- when applied to nonlinear systems. coordinate basin is shown in blue, the minimal-coordinate basin in red, and the basin for both in green.
    [Show full text]