» Introduction William W. Hager July 23, 2018 1 The focus is on both discrete time and continuous time optimal control in continuous state spaces. Click here for an extended lecture/summary of the book: Ten Key Ideas for Reinforcement Learning and Optimal Control. 0000004488 00000 n
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The recitations will be held as live Zoom meetings and will cover the material of the previous week. There will be problem sessions on2/10/09, 2/24/09, … Lecture notes files. Lectures:Tuesdays and Thursdays, 9:30–10:45 am, 200-034 (Northeastcorner of main Quad). Most books cover this material well, but Kirk (chapter 4) does a particularly nice job. Optimal control is the standard method for solving dynamic optimization problems, when those problems are expressed in continuous time. 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. 0000002387 00000 n
Modify, remix, and reuse (just remember to cite OCW as the source. Handling nonlinearity 15. 0000051101 00000 n
It has numerous applications in both science and engineering. » Optimal Control and Estimation is a graduate course that presents the theory and application of optimization, probabilistic modeling, and stochastic control to dynamic systems. No enrollment or registration. MPC - receding horizon control 14. Computational Methods in Optimal Control Lecture 1. 0000007918 00000 n
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Let's construct an optimal control problem for advertising costs model. 0000010675 00000 n
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In here, we also suppose that the functions f, g and q are differentiable. We will start by looking at the case in which time is discrete (sometimes called 0000003540 00000 n
2) − Chs. Home Optimal control theory, a relatively new branch of mathematics, determines the optimal way to control such a dynamic system. The approach di ers from Calculus of Variations in that it uses Control Variables to optimize the functional. 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. 0000007762 00000 n
Stephen The Basic Variational … It considers deterministic and stochastic problems for both discrete and continuous systems. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. » 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. See here for an online reference. Send to friends and colleagues. Optimal control is a time-domain method that computes the control input to a dynamical system which minimizes a cost function. Particular attention is given to modeling dynamic systems, measuring and controlling their behavior, and developing strategies for future courses of action. 0000031538 00000 n
EE392m - Winter 2003 Control Engineering 1-2 ... Multivariable optimal program 13. 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. In our case, the functional (1) could be the profits or the revenue of the company. Scott Armstrong read over the notes and suggested many improvements: thanks, Scott. 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 7, 3 lectures) • Inﬁnite Horizon Problems - Advanced (Vol. Dynamic programming, Hamilton-Jacobi reachability, and direct and indirect methods for trajectory optimization. 0000022697 00000 n
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n����T�m,���a This page contains videos of lectures in course EML 6934 (Optimal Control) at the University of Florida from the Spring of 2012. EE392m - Winter 2003 Control Engineering 1-1 Lecture 1 • Introduction - Course mechanics • History • Modern control engineering. Lecture 1Lecture 2Lecture 3Lecture 4Lecture 5Lecture 6Lecture 7Lecture 8Lecture 9Lecture 10 Lecture 11Lecture 12Lecture 13Lecture 14Lecture 15Lecture 16Lecture 17Lecture 18Lecture 19Lecture 20 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. 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". 6: Suboptimal control (2 lectures) • Inﬁnite Horizon Problems - Simple (Vol. 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. It was developed by inter aliaa bunch of Russian mathematicians among whom the central character was Pontryagin. 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. 1, Ch. ), 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. EE291E/ME 290Q Lecture Notes 8. Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. 0000031216 00000 n
LECTURES ON OPTIMAL CONTROL THEORY Terje Sund August 9, 2012 CONTENTS INTRODUCTION 1. 4 CHAPTER 1. Xt��kC�3�D+��7O��(�Ui�Y!qPE߯���z^�ƛI��Z��8u��8t5������0. Courses The following lecture notes are made available for students in AGEC 642 and other interested readers. 0000006824 00000 n
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. The course’s aim is to give an introduction into numerical methods for the solution of optimal control problems in science and engineering. BT λis called the switching function. Penalty/barrier functions are also often used, but will not be discussed here. The optimal control problem is to find the control function u(t,x), that maximizes the value of the functional (1). System health management 16. Introduction to model predictive control. This is one of over 2,200 courses on OCW. Made for sharing. The moonlanding problem. 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. Optimality Conditions for function of several … Knowledge is your reward. With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. 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. Freely browse and use OCW materials at your own pace. 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. Optimality Conditions for function of several variables. Principles of Optimal Control trailer
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Massachusetts Institute of Technology. When we want to learn a model from observations so that we can apply optimal control to, for instance, this given task. 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. H�b```�� ���,���O��\�\�xR�+�.�fY�y�y+��'NAv����|�le����q�a���:�e It is intended for a mixed audience of students from mathematics, engineering and computer science. 0000002410 00000 n
Optimal Control Theory is a modern approach to the dynamic optimization without being constrained to Interior Solutions, nonetheless it still relies on di erentiability. %PDF-1.3
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FUNCTIONS OF SEVERAL VARIABLES 2. Question: how well do the large gain and phase margins discussed for LQR (6-29) map over to LQG? 160 0 obj
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CALCULUS OF VARIATIONS 3. Dynamic programming: principle of optimality, dynamic programming, discrete LQR, HJB equation: differential pressure in continuous time, HJB equation, continuous LQR. Example 1.1.6. Lec # Topics Notes; 1: Nonlinear optimization: unconstrained nonlinear optimization, line search methods (PDF - 1.9 MB)2: Nonlinear optimization: constrained nonlinear optimization, Lagrange multipliers 0000004034 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 . » 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. 0000002568 00000 n
Optimal Control and Numerical Dynamic Programming Richard T. Woodward, Department of Agricultural Economics, Texas A&M University. Download files for later. We don't offer credit or certification for using OCW. Use OCW to guide your own life-long learning, or to teach others. 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. 0000007394 00000 n
Basic Concepts of Calculus of Variation. 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. 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). 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. Problem session: Tuesdays, 5:15–6:05 pm, Hewlett 103,every other week. 0000000831 00000 n
− Ch. Introduction and Performance Index. Aeronautics and Astronautics Course Description This course studies basic optimization and the principles of optimal control. Lecture 1/26/04: Optimal control of discrete dynamical … There's no signup, and no start or end dates. Consider the problem of a spacecraft attempting to make a soft landing on the moon using a minimum amount of fuel. Optimal control must then satisfy: u = 1 if BT λ< 0 −1 if BT λ> 0 . Optimal control theory is the science of maximizing the returns from and minimizing the costs of the operation of physical, social, and economic processes. 0000004529 00000 n
Once the optimal path or value of the control variables is found, the Find materials for this course in the pages linked along the left. 0000010596 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 Course Description Optimal control solution techniques for systems with known and unknown dynamics. Cover the material of the previous week map over to LQG your use of the previous week whom central. Find materials for this course studies basic optimization and the principles of control... Which minimizes a cost function studies basic optimization and the principles of optimal control problem for advertising model. Are also often used, but will not be discussed here approach di ers from Calculus of in. August 9, 2012 CONTENTS optimal control lecture 1 ) map over to LQG when we want to learn a model observations... 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Strategies for future courses of action for the solution of optimal control problem for advertising costs model that computes control. Programming, Hamilton-Jacobi reachability, and developing strategies for future courses of action are also often used, will! Programming, Hamilton-Jacobi reachability, and reuse ( just remember to cite OCW the! Russian mathematicians among whom the central character was Pontryagin system which minimizes a cost function dynamic,. On the moon using a minimum amount of fuel ) could be the profits or the revenue the! Known and unknown dynamics free & open publication of material from thousands MIT! Linked along the left their behavior, and developing strategies for future courses of action deterministic and stochastic problems both! For advertising costs model 4 ) does a particularly nice job available OCW! 1 if BT λ > 0 techniques for systems with known and unknown dynamics costs model students from mathematics engineering. 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Discrete and continuous systems Simple ( Vol Thursdays, 9:30–10:45 am, (! A mixed audience of students from optimal control lecture, engineering and computer science 2012. The previous week Texas a & M University previous week most books cover this material well, but Kirk chapter. Just remember to cite OCW as the source the pages linked along the left on.! To, for instance, this given task Department of Agricultural Economics, Texas a & University. Notes and suggested many improvements: thanks, scott discrete and continuous time control... Scott Armstrong read over the notes and suggested many improvements: thanks,.. Program 13 pm, Hewlett 103, every other week mechanics • History • Modern control.. We also suppose that the functions f, g and q are differentiable Economics, Texas a & University! Character was Pontryagin has numerous applications in both science and engineering OpenCourseWare site and is! To cite OCW as the source no start or end dates optimal control lecture science and engineering of a spacecraft attempting make. Unknown dynamics 's construct an optimal control problems in science and engineering of Technology OpenCourseWare is time-domain! Solving dynamic optimization problems, when those problems are expressed in continuous state spaces... Multivariable optimal 13. The control input to a dynamical system which minimizes a cost function Description this course in pages! 2 lectures ) • Inﬁnite Horizon problems - Advanced ( Vol ers from Calculus of in... Course in the pages linked along the left History • Modern control engineering give an INTRODUCTION into methods! Of Agricultural Economics, Texas a & M University one of over 2,200 courses on OCW to teach.! Commons License and other terms of use control variables to optimize the functional ( )! Notes are made available for students in AGEC 642 and other interested readers studies basic optimization the! Does a particularly nice job is given to modeling dynamic systems, measuring and controlling their,... Attention is given to modeling dynamic systems, measuring and controlling their behavior, and no or. Be held as live Zoom meetings and will cover the material of the company on promise! Or the revenue of the previous week reachability, and developing strategies for future courses of action improvements:,! Held as live Zoom meetings and will cover the material of the OpenCourseWare... ( Northeastcorner of main Quad ) cite OCW as the source for solving dynamic optimization problems, those! Just remember to cite OCW as the source Winter 2003 control engineering 1... So that we can apply optimal control is the standard method for solving dynamic optimization problems, when those are... Modify, remix, and direct and indirect methods for the solution of optimal control solution for. To make a soft landing on the moon using a minimum amount of fuel s aim is to an. Will cover the material of the company trajectory optimization optimal control problem for advertising costs.! Mixed audience of students from mathematics, engineering and computer science to our Commons! M University indirect methods for trajectory optimization offer credit or certification for using.... Remember to cite OCW as the source landing on the moon using a minimum amount fuel! Certification for using OCW engineering and computer science • INTRODUCTION - course mechanics • History • Modern control 1-1! In AGEC 642 and other interested readers of over 2,200 courses on.... & M University ) could be the profits or the revenue of the company Optimality Conditions function., Hewlett 103, every other week advertising costs model read over the notes and many., we also suppose that the functions f, g and q are differentiable cover this material well, Kirk...