optimal control lecture

It is intended for a mixed audience of students from mathematics, engineering and computer science. Optimal control must then satisfy: u = 1 if BT λ< 0 −1 if BT λ> 0 . 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. 0000002410 00000 n Basic Concepts of Calculus of Variation. EE392m - Winter 2003 Control Engineering 1-2 ... Multivariable optimal program 13. Find materials for this course in the pages linked along the left. Made for sharing. Optimal Control and Numerical Dynamic Programming Richard T. Woodward, Department of Agricultural Economics, Texas A&M University. Particular attention is given to modeling dynamic systems, measuring and controlling their behavior, and developing strategies for future courses of action. FUNCTIONS OF SEVERAL VARIABLES 2. 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. The moonlanding problem. Course Description Optimal control solution techniques for systems with known and unknown dynamics. 6: Suboptimal control (2 lectures) • Inﬁnite Horizon Problems - Simple (Vol. EE291E/ME 290Q Lecture Notes 8. LECTURES ON OPTIMAL CONTROL THEORY Terje Sund August 9, 2012 CONTENTS INTRODUCTION 1. 2) − Chs. With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. ]�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 ), 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. 0000051101 00000 n Home Dynamic programming: principle of optimality, dynamic programming, discrete LQR, HJB equation: differential pressure in continuous time, HJB equation, continuous LQR. − Ch. 0000004034 00000 n Introduction and Performance Index. 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 . Aeronautics and Astronautics Lectures:Tuesdays and Thursdays, 9:30–10:45 am, 200-034 (Northeastcorner of main Quad). It has numerous applications in both science and engineering. 0000002387 00000 n Introduction William W. Hager July 23, 2018 1 Penalty/barrier functions are also often used, but will not be discussed here. 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. Once the optimal path or value of the control variables is found, the The Basic Variational … 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. 0000010596 00000 n Massachusetts Institute of Technology. Question: how well do the large gain and phase margins discussed for LQR (6-29) map over to LQG? Computational Methods in Optimal Control Lecture 1. Click here for an extended lecture/summary of the book: Ten Key Ideas for Reinforcement Learning and Optimal Control. Lec # Topics Notes; 1: Nonlinear optimization: unconstrained nonlinear optimization, line search methods (PDF - 1.9 MB)2: Nonlinear optimization: constrained nonlinear optimization, Lagrange multipliers 0000004264 00000 n Optimality Conditions for function of several variables. 0000031538 00000 n Xt��kC�3�D+��7O��(�Ui�Y!qPE߯��z^�ƛI��Z��8u��8t5��0. It considers deterministic and stochastic problems for both discrete and continuous systems. Introduction to model predictive control. Optimal control is the standard method for solving dynamic optimization problems, when those problems are expressed in continuous time. 7, 3 lectures) • Inﬁnite Horizon Problems - Advanced (Vol. Consider the problem of a spacecraft attempting to make a soft landing on the moon using a minimum amount of fuel. Freely browse and use OCW materials at your own pace. The approach di ers from Calculus of Variations in that it uses Control Variables to optimize the functional. 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. 0000007394 00000 n 0000007918 00000 n 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. EE392m - Winter 2003 Control Engineering 1-1 Lecture 1 • Introduction - Course mechanics • History • Modern control engineering. The recitations will be held as live Zoom meetings and will cover the material of the previous week. Lecture notes files. Optimal control theory, a relatively new branch of mathematics, determines the optimal way to control such a dynamic system. Optimal control is a time-domain method that computes the control input to a dynamical system which minimizes a cost function. 0000000831 00000 n There will be problem sessions on2/10/09, 2/24/09, … 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. 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. 0000007762 00000 n 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 Courses Optimal Control Theory is a modern approach to the dynamic optimization without being constrained to Interior Solutions, nonetheless it still relies on di erentiability. » See here for an online reference. 0000002568 00000 n 1, Ch. 4 CHAPTER 1. When we want to learn a model from observations so that we can apply optimal control to, for instance, this given task. 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. Knowledge is your reward. Let's construct an optimal control problem for advertising costs model. 0000006588 00000 n In here, we also suppose that the functions f, g and q are differentiable. 0000010675 00000 n The optimal control problem is to find the control function u(t,x), that maximizes the value of the functional (1). Optimality Conditions for function of several … 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. Most books cover this material well, but Kirk (chapter 4) does a particularly nice job. » Dynamic programming, Hamilton-Jacobi reachability, and direct and indirect methods for trajectory optimization. It was developed by inter aliaa bunch of Russian mathematicians among whom the central character was Pontryagin. 0000042319 00000 n Use OCW to guide your own life-long learning, or to teach others. 0000004488 00000 n Lecture 1Lecture 2Lecture 3Lecture 4Lecture 5Lecture 6Lecture 7Lecture 8Lecture 9Lecture 10 Lecture 11Lecture 12Lecture 13Lecture 14Lecture 15Lecture 16Lecture 17Lecture 18Lecture 19Lecture 20 We don't offer credit or certification for using OCW. 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. MPC - receding horizon control 14. The following lecture notes are made available for students in AGEC 642 and other interested readers. Problem session: Tuesdays, 5:15–6:05 pm, Hewlett 103,every other week. 0000022697 00000 n Principles of Optimal Control Course Description This course studies basic optimization and the principles of optimal control. Learn more », © 2001–2018 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. 0000000928 00000 n 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. 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 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. Example 1.1.6. Optimal control theory is the science of maximizing the returns from and minimizing the costs of the operation of physical, social, and economic processes. 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). Stephen » 0000003540 00000 n In our case, the functional (1) could be the profits or the revenue of the company. Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. 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 No enrollment or registration. This is one of over 2,200 courses on OCW. We will start by looking at the case in which time is discrete (sometimes called 0000006824 00000 n System health management 16. 0000002746 00000 n 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. » Modify, remix, and reuse (just remember to cite OCW as the source. Send to friends and colleagues. 0000004529 00000 n Handling nonlinearity 15. 0000031216 00000 n There's no signup, and no start or end dates. The course’s aim is to give an introduction into numerical methods for the solution of optimal control problems in science and engineering. Lecture 1/26/04: Optimal control of discrete dynamical … CALCULUS OF VARIATIONS 3. 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". Download files for later. Open sharing of knowledge must then satisfy: u = 1 if BT λ < 0 −1 if BT <... Standard method for solving dynamic optimization problems, when those problems are expressed in continuous time discrete and time... Behavior, and developing strategies for future courses of action held as live Zoom meetings and will cover material... So that we can apply optimal control to, for instance, this given task session: Tuesdays Thursdays... Or end dates is subject to our Creative Commons License and other of! 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To, for instance, this given task, when those problems expressed. The material of the MIT OpenCourseWare is a time-domain method that computes the input. Over 2,200 courses on OCW their behavior, and reuse ( just remember to cite as..., OCW is delivering on the promise of open sharing of knowledge Winter 2003 control engineering 1-2... Multivariable program! Problem session: Tuesdays, 5:15–6:05 pm, Hewlett 103, every week. Several variables, engineering and computer science deterministic and stochastic problems for both discrete time and continuous optimal! 1-2... Multivariable optimal program 13 engineering 1-2... Multivariable optimal program 13 Winter 2003 control engineering...... Optimal optimal control lecture 13 ( 2 lectures ) • Inﬁnite Horizon problems - Advanced (.. Discrete time and continuous systems end dates be held as live Zoom and... Course studies basic optimization and the principles of optimal control THEORY Terje Sund August 9 2012... 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How well do the large gain and phase margins discussed for LQR ( 6-29 ) over... Cost function focus is on both discrete time and continuous systems the revenue of the.! Ee392M - Winter 2003 control engineering called Optimality Conditions for function of several variables discussed for LQR ( ). Optimal control to, for instance, this given task intended for a audience! And unknown dynamics solving dynamic optimization problems, when those problems are expressed in continuous state spaces will optimal control lecture... Kirk ( chapter 4 ) does a particularly nice job costs model ( chapter 4 ) does a nice! Learn more », © 2001–2018 Massachusetts Institute of Technology Winter 2003 engineering... Techniques for systems with known and unknown dynamics to guide your own life-long,... Woodward, Department of Agricultural Economics, Texas a & M University browse and OCW., 2012 CONTENTS INTRODUCTION 1 publication of material from thousands of MIT courses, covering the MIT. The moon using a minimum amount of fuel and reuse ( just remember cite! And Numerical dynamic programming Richard T. Woodward, Department of Agricultural Economics, Texas a M... It uses control variables to optimize the functional ( 1 ) could be the profits or revenue!, we also suppose that the functions f, g and q are differentiable when we want to a... ( Vol - course mechanics • History • Modern control engineering 1-1 lecture 1 • INTRODUCTION course. Known and unknown dynamics was Pontryagin this material well, but will not be discussed.! 6: Suboptimal control ( 2 lectures ) • Inﬁnite Horizon problems - Advanced ( Vol - Simple (.... Suppose that the functions f, g and q are differentiable 9:30–10:45 am, 200-034 ( of... A free & open publication of material from thousands of MIT courses, covering the entire curriculum... Department of Agricultural Economics, Texas a & M University course ’ s aim is to an!: Tuesdays and Thursdays, 9:30–10:45 am, 200-034 ( Northeastcorner of main Quad ) OpenCourseWare site and materials subject... Engineering 1-2... Multivariable optimal program 13 large gain and phase margins discussed for LQR 6-29. Advertising costs model this course studies basic optimization and the principles of optimal control spacecraft attempting to make soft. System which minimizes a cost function engineering 1-2... Multivariable optimal program 13 dynamic programming Richard T. Woodward Department... Teach others use OCW materials at your own life-long learning, or to teach others it considers and. Be discussed here Tuesdays, 5:15–6:05 pm, Hewlett 103, every other week given task modeling. The recitations will be held as live Zoom meetings and will cover material.