# dynamic programming and optimal control course

Markov chains; linear programming; mathematical maturity (this is a doctoral course). Based on Chapters 1 and 6 of the book Dynamic Programming and Optimal Control, Vol. 3 Units. This course serves as an advanced introduction to dynamic programming and optimal control. The second one that we can use is called the maximum principle or the Pontryagin's maximum principle, but we will use the first one. Dynamic Programming and Optimal Control by Dimitri P. Bertsekas, Vol. For that, we can use the so-called Bellman equation which is presented below in the slide. Dynamic Programming and Optimal Control Fall 2009 Problem Set: Deterministic Continuous-Time Optimal Control Notes: • Problems marked with BERTSEKAS are taken from the book Dynamic Programming and Optimal Control by Dimitri P. Bertsekas, Vol. The course covers the basic models and solution techniques for problems of sequential decision making under uncertainty (stochastic control). Let's suppose that the Bellman function has the form presented on the slide or let's try to define it in this particular form. Download Ebook Dynamic Programming And Optimal Control ... Click here to download lecture slides for the MIT course "Dynamic Programming and Stochastic Control (6.231), Dec. 2015. Dynamic Programming and Optimal Control Preface: This two-volume book is based on a first-year graduate course on dynamic programming and optimal control that I have taught for over twenty years at Stanford University, the University of Illinois, and the Massachusetts Institute of Technology. We pay special attention to the contexts of dynamic programming/policy iteration and control theory/model predictive control. supports HTML5 video. But in order to get more information about that you can look at the list of references. Because the Bellman function can be any continuously differentiable function. Dynamic Programming and Optimal Control 3rd Edition, Volume II by Dimitri P. Bertsekas Massachusetts Institute of Technology Chapter 6 Approximate Dynamic Programming Interchange arguments and optimality of index policies in multi-armed bandits and control of queues. Grading The final exam covers all material taught during the course, i.e. The formula (4) defines the differential equation or the motion equation for this dynamical system. Let's denote the optimal control as a u*(t,x), and the corresponding trajectory as x*(t). According to this statement, we can define the procedure to find the optimal solution of the control problem. What if one is cooperative and the other is not? As a result, the control function will also be calculated using the numerical methods and Bellman function as well. Notation for state-structured models. The second part of the course covers algorithms, treating foundations of approximate dynamic programming and reinforcement learning alongside exact dynamic programming algorithms. So, that the cooperation would be beneficial for all of the participants. The third part is devoted to the topic of cooperative differential games, where the question is of how to allocate the maximum joint payoff of players in the game. Time-varying and periodic systems. Let's suppose that we have a dynamical system. Every day, almost every minute we make a choice. 2. Amazon.in - Buy Dynamic Programming and Optimal Control: 2 book online at best prices in India on Amazon.in. 1 Dynamic Programming Dynamic programming and the principle of optimality. The functions B(t) and A(t) are not known. This course provides basic solution techniques for optimal control and dynamic optimization problems, such as those found in work with rockets, robotic arms, autonomous cars, option pricing, and macroeconomics. Linear estimation and the Kalman filter. Then, as a result, the trajectory of the system or the function x(t) is different. We do not know yet the optimal control. But the question is of how to find the optimal control or how to find a function u(t,x), that would maximize the functional (3). We need to simplify the left and the right-hand side of the Bellman equation so that on the left we will have derivative of the function A(t) multiplied by x plus derivative of the function B(t). In order to do that, we can use a several classical approaches. It is important that we define the strategy as a function of any vertex. How can we do that? The optimal control problem is to find the control function u(t,x), that maximizes the value of the functional (1). The first one is dynamic programming principle or the Bellman equation. 529-552, Dec. 1971. 13th Lecture Course, Arizona State University, 2019 Video on Approximate Dynamic Programming. Also, I did not mention that before but the exponenta^(-rt) defines the discount factor. Schedule: Winter 2020, Mondays 2:30pm - 5:45pm. Then, we can say that the derivative of the function A(t) is equal to the first term that is multiplied by x, and then the derivative of function B(t) is equal to the term on the right-hand side of the Bellman equation. With maximum principle, we can find a solution for a much wider class of problems, but it is only the necessary condition. We define a strategy or a control function u(t,x) for any time instant t and for any state x(t). dynamic programming and optimal control 2 vol set. Choices can be insignificant: to go by tram or by bus, to take an umbrella or not. 1.1 Control as optimization over time Optimization is a key tool in modelling. I will follow the following weighting: 20% homework, 15% lecture scribing, 65% final or course project. When are long-term stable prospects better than short-term benefits, and when not? In order to find functions A(t) and B(t), we need to transform the Bellman equation into the system of differential equations. We also discuss in some detail the application of the methodology to challenging discrete/combinatorial optimization problems, such as routing, scheduling, assignment, and mixed integer programming, including the use of neural network approximations within these contexts. In the same way, we do it in here. Base-stock and (s,S) policies in inventory control, Linear policies in linear quadratic control, Separation principle and Kalman filtering in LQ control with partial observability. The right-hand sign of this differential equation depends on the market share in the current time instant, and also depends on the marketing expenses. Then, we can solve it and define the trajectory x*(t) along which the system or the company would go. What we can do is we can use the numerical methods. dynamic programming and optimal control … In game Ñtheory, we call it the choice of strategy. But in this particular model, the system of differential equations for the functions A(t) and B(t) cannot be solved analytically. The first part of the course will cover problem formulation and problem specific solution ideas arising in canonical control problems. Its revenue mainly depends on the market share. Then the question is, of how it would allocate the advertising costs, when company need to spend more money on advertising and when not. In our case, the functional (1) could be the profits or the revenue of the company. Optimal control solution techniques for systems with known and unknown dynamics. Because in the differential games, this is the approach that is more widely used. The main deliverable will be either a project writeup or a take home exam. Read Dynamic Programming and Optimal Control: 2 book reviews & author details and more at Amazon.in. To view this video please enable JavaScript, and consider upgrading to a web browser that ECE 553 - Optimal Control, Spring 2008, ECE, University of Illinois at Urbana-Champaign, Yi Ma ; U. Washington, Todorov; MIT: 6.231 Dynamic Programming and Stochastic Control Fall 2008 See Dynamic Programming and Optimal Control/Approximate Dynamic Programming, for Fall 2009 course … For Class 2 (2/3): Vol 1 sections 3.1, 3.2. Anyway, if we solve the system of differential equations, we substitute the functions A(t) and B(t) into the optimal control, then we substitute into the Bellman function, then the optimal control as a function of (t,x) we substitute to the motion equation. On this slide you can see a list of references from where you could find more information of how to use the dynamic programming principle, where we could find information about the maximum principle and to find more examples. Well, how can we use that in order to find the optimal control problem (1),(2)? So, if there is a solution for a Bellman equation, then we say that our solution is optimal. References Textbooks, Course Material, Tutorials [Ath71] M. Athans, The role and use of the stochastic linear-quadratic-Gaussian problem in control system design, IEEE Transactions on Automatic Control, 16-6, pp. In several sections, definitions and theorems from mathematical analysis and elements of probability theory will be used. Mathematical Optimization. This course serves as an advanced introduction to dynamic programming and optimal control. On the slide on the right-hand side you can see the optimal control along the corresponding optimal trajectory when x at time instant t is equal to x*(t) and on the left-hand side you can see the corresponding optimal trajectory. Let's construct an optimal control problem for advertising costs model. The solution of this motion equation is the function x(t), which defines the state of the game. This is what you see on the slide. Why? Dynamic Programming and Optimal Control by Dimitris Bertsekas, 4th Edition, Volumes I and II. In our case, under the state of the game, we can understand the market share of the company. But if it is not the linear quadratic game, then on the first step we need to try to find the form for the Bellman function, then we need to try to solve the system of differential equation. For example, specify the state space, the cost functions at each state, etc. So, the functions that depend on t - the time instant and x - the state of the game. There will be a few homework questions each week, mostly drawn from the Bertsekas books. Lyapunov theory and methods. 151-0563-01 Dynamic Programming and Optimal Control (Fall 2020) Class Website All information concerning the class: announcements, class facts, problem sets, etc. Learn Dynamic Programming online with courses like Algorithms and Greedy Algorithms, Minimum Spanning Trees, and Dynamic Programming. QA402.5 .13465 2005 … For Class 3 (2/10): Vol 1 sections 4.2-4.3, Vol 2, sections 1.1, 1.2, 1.4, For Class 4 (2/17): Vol 2 section 1.4, 1.5. So, for any function u(t,x) or for any advertising expenses, the right-hand side of the system of differential equations is different. In order to do that, we need to define the notion of the Bellman function. For the control function u, we will consider a class of functions u(t,x). We will consider optimal control of a dynamical system over both a finite and an infinite number of stages. Free delivery on qualified orders. Who is interested in world politics and at least once heard about the "Prisoner's Dilemma". When we consider a multi-stage game with perfect information, the strategy of the player was a mapping that for each vertex from the set of personal moves of the player i, assigns the next vertex on the graph. You will be asked to scribe lecture notes of high quality. In here, we also suppose that the functions f, g and q are differentiable. I, 4th Edition, Athena Scientific. We will have a short homework each week. In here, we also suppose that the functions f, g and q are differentiable. Examples and applications from digital filters, circuits, signal processing, and control systems. Brief overview of average cost and indefinite horizon problems. Why do those who have agreed to cooperate, suddenly break the agreement? Then after the defining of the control that maximizes the right-hand side, we can derive the optimal control, which is presented on the slide. I, 3rd edition, 2005, 558 pages. The first part of the course will cover problem formulation and problem specific solution ideas arising in canonical control problems. What is the Bellman function? In the similar way, we already defined the control function or the strategy of the players in one of the previous sections. An introductory (video)lecture on dynamic programming within a course on "Optimal and Robust Control" (B3M35ORR, BE3M35ORR, BEM35ORC) … Dynamic Programming courses from top universities and industry leaders. Constantly interacting with society and adopting certain strategies, many of us wonder: why can't everyone exist peacefully and cooperate with each other? Of course, we cannot do that for a general class of problems. In order to construct a mathematical model for this process, the first thing we need to do is to define the optimal control problem. But, we do not know the function A(t). Exam Final exam during the examination session. So, what is the dynamic programming principle? Dynamic programming and optimal control course project involving Policy Iteration, Value Iteration, and Linear Programming - duyipai/Dynamic-Programming-and-Optimal-Control-Project 6-lecture, 12-hour short course, Tsinghua University, Beijing, China, 2014 Lecture slides for a 6-lecture short course on approximate dynamic programming… We say that if there exists a continuously differentiable function V(t,x) satisfying the Bellman equation presented below, which is a partial differential equation, then the function u(t) which maximizes the right-hand side of the Bellman equation is an optimal control in the problem (1),(2). [Bel57] R.E. So, we would need to check the solution once again and prove that it is sufficient. Because the Bellman equation is a sufficient condition for the optimal control. Suppose that we know the optimal control in the problem defined on the interval [t0,T]. strengthen learning and optimal control. Download Course Materials; Unless otherwise indicated, homework problems were taken from the course textbook: Bertsekas, Dimitri P. Dynamic Programming and Optimal Control, Volume I. The Bellman function is a function V(t,x). Course Syllabus: Dynamic Programming and Optimal Control - EE 372 Division Computer, Electrical and Mathematical Sciences & Engineering Course Number EE 372 Course Title Dynamic Programming and Optimal Control Academic Semester Fall Academic Year 2017/2018 Semester Start Date 08/20/2017 Semester End Date 12/12/2017 Class Schedule (Days & Time) But there is a approach that we can use and let's demonstrate it on the advertising costs example. This is true for any truncated interval. The first part is devoted to the study of some preliminary information or the approaches of how to solve differential games. This course will be useful for those who want to make choices based on mathematical calculations rather than relying on fate. Let's suppose that the company wants to make a plan for advertising for one year. Introduction to model predictive control. Sometimes they can be very significant and even crucial: the choice of University, life partner. Realization theory. 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 But, there is a problem. As you can see, it does not depend on the optimal control or on any control, because the value of the Bellman function is already optimal. L Title. Lectures on Exact and Approximate Infinite Horizon DP: Videos from a 6-lecture, 12-hour short course at Tsinghua Univ. In the position on the optimal trajectory. A Short Proof of the Gittins Index Theorem, Connections between Gittins Indices and UCB, slides on priority policies in scheduling, Partially observable problems and the belief state. 3rd ed. Athena Scientific, 2005. The right hand side of the system of differential equations, also depends on the function u(t), which is a control function or, in our case, the advertising costs. So, for if we fix the market share of the company, and then we try to change the advertising costs, then of course the advertising costs are higher than the value of the functional is lower. similarities and differences between stochastic. Foundations of reinforcement learning and approximate dynamic programming. But, for example, for a linear quadratic games the explicit solution is known. Lectures in Dynamic Programming and Stochastic Control Arthur F. Veinott, Jr. Spring 2008 ... Optimal Control of Tandem Queues Homework 6 (5/16/08) ... yond the finite horizon—which they might view as speculative anyway—though of course these pro- In our case, it is a company. Dynamic Programming and Optimal Control Includes Bibliography and Index 1. Why? On the right-hand side, we're going to have a term multiplied by x plus some other term. I, 3rd edition, 2005, 558 pages, hardcover. Dynamic Programming and Optimal Control is offered within DMAVT and attracts in excess of 300 students per year from a wide variety of disciplines. Exact algorithms for problems with tractable state-spaces. For a different function x(t), and for different control function, we have the different, values of the functional (1). This includes systems with finite or infinite state spaces, as well as perfectly or imperfectly observed systems. It will be periodically updated as dynamic programming and optimal control 3rd edition volume ii. In this section, a neuro-dynamic programming algorithm is developed to solve the constrained optimal control problem. The answers to these and other questions you will find out in our course. So, of how people discount the payoffs that they are going to obtain in the future. The choice may affect a small group of people or entire countries. We cannot solve the Bellman equation for a general class of problems. I, 3rd edition, 2005, 558 pages, hardcover. Construction Engineering and Management Certificate, Machine Learning for Analytics Certificate, Innovation Management & Entrepreneurship Certificate, Sustainabaility and Development Certificate, Spatial Data Analysis and Visualization Certificate, Master's of Innovation & Entrepreneurship. The dynamics of the system is defined by the system of differential equations or motion equations (2). The course is in part based on a tutorial given at ICML 2008 and on some selected material from the book Dynamic programming and optimal control by Dimitri Bertsekas. Short course on control theory and dynamic programming - Madrid, October 2010 The course provides an introduction to stochastic optimal The course is in part based on a tutorial given by me and Marc Toussaint at ICML 2008 and on some selected material from the book Dynamic programming and optimal control by Dimitri Bertsekas. On the market, there is a company who tries to maximize its revenue. Hello. Optimal control and dynamic programming; linear quadratic regulator. Model-based reinforcement learning, and connections between modern reinforcement learning in continuous spaces and fundamental optimal control ideas. Bellman, "Dynamic Programming", Dover, 2003 [Ber07] D.P. Short course on control theory and dynamic programming - Madrid, January 2012 The course provides an introduction to stochastic optimal control theory. If we can solve the Bellman equation, then the corresponding control would be optimal. So, the company can control the advertising costs. Then, the partial derivatives would be the derivatives of the functions A(t) and B(t). Please write down a precise, rigorous, formulation of all word problems. Then, the truncation of the optimal control u*(t,x) on the subproblem defined on the interval [t',T], would be also optimal in the problem starting at time instant t' and in the position x*(t'). Dynamic Programming. The optimal control problem is to find the control function u(t,x), that maximizes the value of the functional (1). Dynamic programming, Hamilton-Jacobi reachability, and direct and indirect methods for trajectory optimization. Due Monday 2/3: Vol I problems 1.23, 1.24 and 3.18. But it has some disadvantages and we will talk about that later. Let's construct an optimal control problem for advertising costs model. Sometimes it is important to solve a problem optimally. The last six lectures cover a lot of the approximate dynamic programming material. Firstly, a neural network is introduced to approximate the value function in Section 4.1, and the solution algorithm for the constrained optimal control based on policy iteration is presented in Section 4.2. On the slide, the formula (3) defines the functional that we need to maximize, which is a revenue of the company on the interval [0,T], which depends on the state of the game or the state function x(t) on this period, and on the advertising expenses. Vol II problems 1.5 and 1.14. Due Monday 4/13: Read Bertsekas Vol II, Section 2.4 Do problems 2.5 and 2.9, For Class 1 (1/27): Vol 1 sections 1.2-1.4, 3.4. The topic of the today's lecture is the differential games. Feedback, open-loop, and closed-loop controls. Let's start with the example called optimization of advertising costs and consider a market. How can we solve a partial differential equation? In our case, the functional (1) could be the profits or the revenue of the company. ISBN: 9781886529267. However, the importance of choice may not be realized initially. How profitable should the interaction be for the opponent to change his opinion? Dynamic Programming and Optimal Control 3rd Edition, Volume II by Dimitri P. Bertsekas Massachusetts Institute of Technology Chapter 6 Approximate Dynamic Programming This is an updated version of the research-oriented Chapter 6 on Approximate Dynamic Programming. Sometimes a decision "not to take an umbrella" radically changes everything. On the slide, you can see the Bellman equation corresponding to the advertising costs problem, and the question is of how to solve it. We need to substitute this form of Bellman function into the Bellman equation. Also, we will suppose that the conditions of existence, uniqueness and prolongability of the system of differential equations (2) for each of such function exist. So, in general, in differential games, people use the dynamic programming principle. It is an integral part of the Robotics, System and Control (RSC) Master Program and almost everyone taking this Master takes this class. To view this video please enable JavaScript, and consider upgrading to a web browser that. The second part is devoted to the non-cooperative differential games of n players, where the main question is of how to model the behavior of players in processes where they have individual preferences or each player has his own payoff function. Course Syllabus: Dynamic Programming and Optimal Control - EE 372 Division Computer, Electrical and Mathematical Sciences & Engineering Course Number EE 372 Course Title Dynamic Programming and Optimal Control Academic Semester Spring Academic Year 2019/2020 Semester Start Date 01/26/2020 Semester End Date 05/13/2020 Class Schedule (Days & Time) on approximate DP, Beijing, China, 2014. Â© 2020 Coursera Inc. All rights reserved. The course is basic and does not require any special knowledge. It's important to know that this is a sufficient condition for the optimal control. Dynamic Programming and Optimal Control Fall 2009 Problem Set: The Dynamic Programming Algorithm Notes: • Problems marked with BERTSEKAS are taken from the book Dynamic Programming and Optimal Control by Dimitri P. Bertsekas, Vol. We also can define the corresponding trajectory. From the Tsinghua course site, and from Youtube. So, when the V(t,x) is equal to exponenta^(-rt) multiplied to sum of function A(t)*x and function B(t). Due Monday 2/17: Vol I problem 4.14 parts (a) and (b). It is the optimal value of the functional (3) defined in the subproblem starting at that time instant t and in the state x(t) or when the initial condition for the motion equation system, differential equation not at zero but at t and x(t). Reinforcement Learning and Optimal Control ASU, CSE 691, Winter 2019 Dimitri P. Bertsekas dimitrib@mit.edu Lecture 1 Bertsekas Reinforcement Learning 1 / 21. The only tool that can be used in order to increase the market share is the advertising. Right now you have made the choice to read this text instead of scrolling further. Markov decision processes. So, it only depends on the initial time instant and state of the subproblem. Dynamic Programming And Optimal Control optimization and control university of cambridge.