0000003540 00000 n Calculus of variations applied to optimal control, Bryson and Ho, Section 3.5 and Kirk, Section 4.4, Bryson and Ho, section 3.x and Kirk, section 5.3, Bryson, chapter 12 and Gelb, Optimal Estimation, Kwaknernaak and Sivan, chapters 3.6, 5; Bryson, chapter 14; and Stengel, chapter 5. Consider the problem of a spacecraft attempting to make a soft landing on the moon using a minimum amount of fuel. Dynamic programming: principle of optimality, dynamic programming, discrete LQR, HJB equation: differential pressure in continuous time, HJB equation, continuous LQR. Lecture 1Lecture 2Lecture 3Lecture 4Lecture 5Lecture 6Lecture 7Lecture 8Lecture 9Lecture 10 Lecture 11Lecture 12Lecture 13Lecture 14Lecture 15Lecture 16Lecture 17Lecture 18Lecture 19Lecture 20 Lecture Notes, LQR = linear-quadratic regulator LQG = linear-quadratic Gaussian HJB = Hamilton-Jacobi-Bellman, Nonlinear optimization: unconstrained nonlinear optimization, line search methods, Nonlinear optimization: constrained nonlinear optimization, Lagrange multipliers. Optimality Conditions for function of several … H�b```�� ���,���O��\�\�xR�+�.�fY�y�y+��'NAv����|�le����q�a���:�e Aeronautics and Astronautics Modify, remix, and reuse (just remember to cite OCW as the source. Lecture notes files. An extended lecture/slides summary of the book Reinforcement Learning and Optimal Control: Ten Key Ideas for Reinforcement Learning and Optimal Control Videolectures on Reinforcement Learning and Optimal Control: Course at Arizona State University , 13 lectures, January-February 2019. 0000051101 00000 n Dynamic Optimization and Optimal Control Mark Dean+ Lecture Notes for Fall 2014 PhD Class - Brown University 1Introduction To finish offthe course, we are going to take a laughably quick look at optimization problems in dynamic settings. The optimal control problem is to find the control function u(t,x), that maximizes the value of the functional (1). When we want to learn a model from observations so that we can apply optimal control to, for instance, this given task. 0000006588 00000 n » Optimality Conditions for function of several variables. Optimal Control and Numerical Dynamic Programming Richard T. Woodward, Department of Agricultural Economics, Texas A&M University. 0000010596 00000 n BT λis called the switching function. Lecture 10 — Optimal Control Introduction Static Optimization with Constraints Optimization with Dynamic Constraints The Maximum Principle Examples Material Lecture slides References to Glad & Ljung, part of Chapter 18 D. Liberzon, Calculus of Variations and Optimal Control Theory: A concise Introduction, Princeton University Download files for later. It is intended for a mixed audience of students from mathematics, engineering and computer science. 0000004264 00000 n 0000042319 00000 n 0000002387 00000 n We don't offer credit or certification for using OCW. LECTURES ON OPTIMAL CONTROL THEORY Terje Sund August 9, 2012 CONTENTS INTRODUCTION 1. Dynamic programming, Hamilton-Jacobi reachability, and direct and indirect methods for trajectory optimization. There will be problem sessions on2/10/09, 2/24/09, … Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. Optimal Control Theory Version 0.2 By Lawrence C. Evans Department of Mathematics University of California, Berkeley ... his notes into a first draft of these lectures as they now appear. The moonlanding problem. Use OCW to guide your own life-long learning, or to teach others. OPTIMAL CONTROL THEORY INTRODUCTION In the theory of mathematical optimization one try to nd maximum or minimum points of functions depending of real variables and of other func-tions. This page contains videos of lectures in course EML 6934 (Optimal Control) at the University of Florida from the Spring of 2012. 16.31 Feedback Control Systems: multiple-input multiple-output (MIMO) systems, singular value decomposition, Signals and system norms: H∞ synthesis, different type of optimal controller. FUNCTIONS OF SEVERAL VARIABLES 2. Freely browse and use OCW materials at your own pace. In our case, the functional (1) could be the profits or the revenue of the company. Basic Concepts of Calculus of Variation. The following lecture notes are made available for students in AGEC 642 and other interested readers. 0000004034 00000 n Deterministic Continuous Time Optimal Control: Slides, Notes: Lecture 9: 10: Dec 02: Pontryagin’s Minimum Principle: Slides, Notes: Lecture 10: 11: Dec 09: Pontryagin’s Minimum Principle (cont’d) Slides, Notes: Lecture 11: Recitations. EE291E/ME 290Q Lecture Notes 8. Let's construct an optimal control problem for advertising costs model. 0000022697 00000 n EE392m - Winter 2003 Control Engineering 1-1 Lecture 1 • Introduction - Course mechanics • History • Modern control engineering. Penalty/barrier functions are also often used, but will not be discussed here. It was developed by inter aliaa bunch of Russian mathematicians among whom the central character was Pontryagin. The dual problem is optimal estimation which computes the estimated states of the system with stochastic disturbances by minimizing the errors between the true states and the estimated states. No enrollment or registration. 0000002746 00000 n Example: Minimum time control of double integrator ¨x = u with specified initial condi-tion x0 and final condition x f = 0, and control constraint |u| ≤ 1. 160 0 obj << /Linearized 1 /O 162 /H [ 928 1482 ] /L 271225 /E 51460 /N 41 /T 267906 >> endobj xref 160 24 0000000016 00000 n Massachusetts Institute of Technology. 0000000831 00000 n In optimal control we will encounter cost functions of two variables L: Rn Rm!R written as L(x;u) where x2R n denotes the state and u2R m denotes the control inputs . 0000007918 00000 n We will start by looking at the case in which time is discrete (sometimes called Optimal control Open-loop Indirect methods Direct methods Closed-loop DP HJB / HJI MPC Adaptive optimal control Model-based RL Linear methods Non-linear methods AA 203 | Lecture 18 LQR iLQR DDP Model-free RL LQR Reachability analysis State/control param Control CoV NOC PMP param 6/8/20 The course’s aim is to give an introduction into numerical methods for the solution of optimal control problems in science and engineering. The Basic Variational … MPC - receding horizon control 14. Principles of Optimal Control There's no signup, and no start or end dates. Introduction and Performance Index. Learn more », © 2001–2018 EE392m - Winter 2003 Control Engineering 1-2 ... Multivariable optimal program 13. trailer << /Size 184 /Info 158 0 R /Root 161 0 R /Prev 267895 /ID[<24a059ced3a02fa30e820d921c33b5e6>] >> startxref 0 %%EOF 161 0 obj << /Type /Catalog /Pages 153 0 R /Metadata 159 0 R /PageLabels 151 0 R >> endobj 182 0 obj << /S 1957 /L 2080 /Filter /FlateDecode /Length 183 0 R >> stream This is one of over 2,200 courses on OCW. 0000000928 00000 n Course Description Optimal control solution techniques for systems with known and unknown dynamics. 0000006824 00000 n Optimal control must then satisfy: u = 1 if BT λ< 0 −1 if BT λ> 0 . Particular attention is given to modeling dynamic systems, measuring and controlling their behavior, and developing strategies for future courses of action. The approach di ers from Calculus of Variations in that it uses Control Variables to optimize the functional. Courses INTRODUCTION TO OPTIMAL CONTROL One of the real problems that inspired and motivated the study of optimal control problems is the next and so called \moonlanding problem". Computational Methods in Optimal Control Lecture 1. g3�,� �%�^�B�Z��m�y�9��"�vi&t�-��ڥ�hZgj��B獹@ԥ�j�b��) �T���^�b�?Q墕����r7}ʞv�q�j��J��P���op{~��b5&�B�0�Dg���d>�/�U ��u'�]�lL�(Ht:��{�+˚I�g̞�k�x0C,��MDGB��ϓ ���{�վH�ud�HgU�;tM4f�Kߗ ���J}B^�X9e$S�]��8�kk~o�Ȅ2k������l�:�Q�tC� �S1pCbQwZ�]G�sn�#:M^Ymi���ܼ�rR�� �`���=bi�/]�8E귚,/�ʢ`.%�Bgind�Z�~W�{�^����o�H�i� ��@�C. The purpose of the book is to consider large and challenging multistage decision problems, which can be solved in principle by dynamic programming and optimal control, but their exact solution is computationally intractable. Lecture 1/26/04: Optimal control of discrete dynamical … Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. Home Introduction William W. Hager July 23, 2018 1 Optimal control theory is the science of maximizing the returns from and minimizing the costs of the operation of physical, social, and economic processes. The focus is on both discrete time and continuous time optimal control in continuous state spaces. It considers deterministic and stochastic problems for both discrete and continuous systems. 4 CHAPTER 1. Example 1.1.6. Xt��kC�3�D+��7O��(�Ui�Y!qPE߯���z^�ƛI��Z��8u��8t5������0. Scott Armstrong read over the notes and suggested many improvements: thanks, Scott. The recitations will be held as live Zoom meetings and will cover the material of the previous week. ), Learn more at Get Started with MIT OpenCourseWare, MIT OpenCourseWare makes the materials used in the teaching of almost all of MIT's subjects available on the Web, free of charge. Optimal control theory, a relatively new branch of mathematics, determines the optimal way to control such a dynamic system. Optimal Control Theory is a modern approach to the dynamic optimization without being constrained to Interior Solutions, nonetheless it still relies on di erentiability. Send to friends and colleagues. Knowledge is your reward. 0000010675 00000 n MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. Course Description This course studies basic optimization and the principles of optimal control. » Lecture 1/15/04: Optimal control of a single-stage discrete time system in-class; Lecture 1/22/04: Optimal control of a multi-stage discrete time system in-class; copies of relevant pages from Frank Lewis. 1, Ch. − Ch. 0000007394 00000 n Problem session: Tuesdays, 5:15–6:05 pm, Hewlett 103,every other week. Lec # Topics Notes; 1: Nonlinear optimization: unconstrained nonlinear optimization, line search methods (PDF - 1.9 MB)2: Nonlinear optimization: constrained nonlinear optimization, Lagrange multipliers See here for an online reference. 7, 3 lectures) • Infinite Horizon Problems - Advanced (Vol. Lectures:Tuesdays and Thursdays, 9:30–10:45 am, 200-034 (Northeastcorner of main Quad). 0000031216 00000 n 0000004488 00000 n Find materials for this course in the pages linked along the left. Optimal control is a time-domain method that computes the control input to a dynamical system which minimizes a cost function. For the rest of this lecture, we're going to use as an example the problem of autonomous helicopter patrol, in this case what's known as a nose-in funnel. With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. » Question: how well do the large gain and phase margins discussed for LQR (6-29) map over to LQG? Made for sharing. Most books cover this material well, but Kirk (chapter 4) does a particularly nice job. Introduction to model predictive control. Optimal Control and Estimation is a graduate course that presents the theory and application of optimization, probabilistic modeling, and stochastic control to dynamic systems. Once the optimal path or value of the control variables is found, the ]�ɶ"��ތߤ�P%U�#H!���d�W[�JM�=���XR�[q�:���1�ѭi��-M�>e��"�.vC�G*�k�X��p:u�Ot�V���w���]F�I�����%@ɣ pZc��Q��2)L�#�:5�R����Ó��K@R��tY��V�F{$3:I,:»k���E?Pe�|~���SѝUBClkiVn��� S��F;�wi�՝ȇ����E�Vn.y,�q�qW4�����D��$��]3��)h�L#yW���Ib[#�E�8�ʥ��N�(Lh�9_���ɉyu��NL �HDV�s�1���f=��x��@����49E�4L)�趍5,��^���6�3f�ʻ�\��!#$,�,��zy�ϼ�N��P���{���&�Op�s�d'���>�hy#e���M֋pGS�!W���=�_��$� n����T�m,���a 6: Suboptimal control (2 lectures) • Infinite Horizon Problems - Simple (Vol. In here, we also suppose that the functions f, g and q are differentiable. Stephen » Optimal control is the standard method for solving dynamic optimization problems, when those problems are expressed in continuous time. %PDF-1.3 %���� Click here for an extended lecture/summary of the book: Ten Key Ideas for Reinforcement Learning and Optimal Control. System health management 16. Handling nonlinearity 15. Introduction to Control Theory Including Optimal Control Nguyen Tan Tien - 2002.5 _____ _____ Chapter 11 Bang-bang Control 53 C.11 Bang-bang Control 11.1 Introduction This chapter deals with the control with restrictions: is bounded and might well be possible to have discontinuities. 0000002568 00000 n CALCULUS OF VARIATIONS 3. 0000007762 00000 n 0000002410 00000 n 2) − Chs. 0000004529 00000 n It has numerous applications in both science and engineering. Optimal Control and Dynamic Games S. S. Sastry REVISED March 29th There exist two main approaches to optimal control and dynamic games: 1. via the Calculus of Variations (making use of the Maximum Principle); 2. via Dynamic Programming (making use of the Principle of Optimality). 0000031538 00000 n And suggested many improvements: thanks, scott OpenCourseWare site and materials is subject to our Creative Commons and! Of students from mathematics, engineering and computer science ) could be the profits or revenue! 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