Note: These are working notes used for a course being taught at MIT. They will be updated throughout the Spring semester. Lecture videos are available on YouTube. Model Systems. Nonlinear Planning and Control. Estimation and Learning.

This book is about building robots that move with speed, efficiency, and grace. I believe that this can only be achieve through a tight coupling between mechanical design, passive dynamics, and nonlinear control synthesis.

Therefore, these notes contain selected material from dynamical systems theory, as well as linear and nonlinear control. These notes also reflect a deep belief in computational algorithms playing an essential role in finding and optimizing solutions to complex dynamics and control problems.

## Table of Contents

Algorithms play an increasingly central role in modern control theory; these days even rigorous mathematicians consider finding convexity in a problem therefore making it amenable to an efficient computational solution almost tantamount to an analytical result. Therefore, the notes necessarily also cover selected material from optimization theory, motion planning, and machine learning.

Although the material in the book comes from many sources, the presentation is targeted very specifically at a handful of robotics problems. Concepts are introduced only when and if they can help progress the capabilities we are trying to develop. Many of the disciplines that I am drawing from are traditionally very rigorous, to the point where the basic ideas can be hard to penetrate for someone that is new to the field.

I've made a conscious effort in these notes to keep a very informal, conversational tone even when introducing these rigorous topics, and to reference the most powerful theorems but only to prove them when that proof would add particular insights without distracting from the mainstream presentation. I hope that the result is a broad but reasonably self-contained and readable manuscript that will be of use to any enthusiastic roboticist.

The material in these notes is organized into a few main parts. Many of these algorithms treat the dynamical system as known and deterministic until the last chapters in this part which introduce stochasticity and robustness. The book closes with an "Appendix" that provides slightly more introduction and references for the main topics used in the course. The order of the chapters was chosen to make the book valuable as a reference.

When teaching the course, however, I take a spiral trajectory through the material, introducing robot dynamics and control problems one at a time, and introducing only the techniques that are required to solve that particular problem.

All of the examples and algorithms in this book, plus many more, are now available as a part of our open-source software project:. Please see the appendix for specific instructions for using along with these notes.Simulation-Based Optimization pp Cite as.

This chapter focuses on a problem of control optimization, in particular the Markov decision problem or process. Our discussions will be at a very elementary level, and we will not attempt to prove any theorems. The central aim of this chapter is to introduce the reader to classical dynamic programming in the context of solving Markov decision problems.

In the next chapter, the same ideas will be presented in the context of simulation-based dynamic programming. The main concepts presented in this chapter are 1 Markov chains, 2 Markov decision problems, 3 semi-Markov decision problems, and 4 classical dynamic programming methods.

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Chapter First Online: 07 August This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, log in to check access. Bellman, The theory of dynamic programming.

Bertsekas, Dynamic Programming and Optimal Control3rd edn. Athena Scientific, Belmont, Google Scholar. Filar, K. Gosavi, A risk-sensitive approach to total productive maintenance.

Automatica 42â€” CrossRef Google Scholar. Gosavi, S. Murray, V. Tirumalasetty, S. Shewade, A budget-sensitive approach to scheduling maintenance in a total productive maintenance TPM program. Hillier, G. Lieberman, Introduction to Operations Research7th edn. Johns Jr. Miller Jr. Wiley, New York, Google Scholar. Shapley, Stochastic games. Taylor, S. White, Dynamic programming, Markov chains, and the method of successive approximations.

Personalised recommendations. Cite chapter How to cite? ENW EndNote. Buy options.Dynamic programming is both a mathematical optimization method and a computer programming method.

### Abstract Dynamic Programming, 2nd Edition, 2018

The method was developed by Richard Bellman in the s and has found applications in numerous fields, from aerospace engineering to economics. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively.

Likewise, in computer science, if a problem can be solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have optimal substructure. If sub-problems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the sub-problems.

In terms of mathematical optimization, dynamic programming usually refers to simplifying a decision by breaking it down into a sequence of decision steps over time. This is done by defining a sequence of value functions V 1V 2The definition of V n y is the value obtained in state y at the last time n. Finally, V 1 at the initial state of the system is the value of the optimal solution.

The optimal values of the decision variables can be recovered, one by one, by tracking back the calculations already performed. The latter obeys the fundamental equation of dynamic programming:. Alternatively, the continuous process can be approximated by a discrete system, which leads to a following recurrence relation analog to the Hamiltonâ€”Jacobiâ€”Bellman equation:.

This functional equation is known as the Bellman equationwhich can be solved for an exact solution of the discrete approximation of the optimization equation. In economics, the objective is generally to maximize rather than minimize some dynamic social welfare function. In Ramsey's problem, this function relates amounts of consumption to levels of utility. Loosely speaking, the planner faces the trade-off between contemporaneous consumption and future consumption via investment in capital stock that is used in productionknown as intertemporal choice.

A discrete approximation to the transition equation of capital is given by. Assume capital cannot be negative. Then the consumer's decision problem can be written as follows:. The dynamic programming approach to solve this problem involves breaking it apart into a sequence of smaller decisions. The value of any quantity of capital at any previous time can be calculated by backward induction using the Bellman equation.

Intuitively, instead of choosing his whole lifetime plan at birth, the consumer can take things one step at a time. To actually solve this problem, we work backwards.

For simplicity, the current level of capital is denoted as k. We see that it is optimal to consume a larger fraction of current wealth as one gets older, finally consuming all remaining wealth in period Tthe last period of life. There are two key attributes that a problem must have in order for dynamic programming to be applicable: optimal substructure and overlapping sub-problems. If a problem can be solved by combining optimal solutions to non-overlapping sub-problems, the strategy is called " divide and conquer " instead.

Optimal substructure means that the solution to a given optimization problem can be obtained by the combination of optimal solutions to its sub-problems. Such optimal substructures are usually described by means of recursion. If p is truly the shortest path, then it can be split into sub-paths p 1 from u to w and p 2 from w to v such that these, in turn, are indeed the shortest paths between the corresponding vertices by the simple cut-and-paste argument described in Introduction to Algorithms.

Hence, one can easily formulate the solution for finding shortest paths in a recursive manner, which is what the Bellmanâ€”Ford algorithm or the Floydâ€”Warshall algorithm does. Overlapping sub-problems means that the space of sub-problems must be small, that is, any recursive algorithm solving the problem should solve the same sub-problems over and over, rather than generating new sub-problems.

Now F 41 is being solved in the recursive sub-trees of both F 43 as well as F Even though the total number of sub-problems is actually small only 43 of themwe end up solving the same problems over and over if we adopt a naive recursive solution such as this.

Dynamic programming takes account of this fact and solves each sub-problem only once.We hope this content on epidemiology, disease modeling, pandemics and vaccines will help in the rapid fight against this global problem. Click on title above or here to access this collection.

This self-contained, practical, entry-level text integrates the basic principles of applied mathematics, applied probability, and computational science for a clear presentation of stochastic processes and control for jump diffusions in continuous time.

The author covers the important problem of controlling these systems and, through the use of a jump calculus construction, discusses the strong role of discontinuous and nonsmooth properties versus random properties in stochastic systems. The book emphasizes modeling and problem solving and presents sample applications in financial engineering and biomedical modeling.

Computational and analytic exercises and examples are included throughout. While classical applied mathematics is used in most of the chapters to set up systematic derivations and essential proofs, the final chapter bridges the gap between the applied and the abstract worlds to give readers an understanding of the more abstract literature on jump diffusions.

The aim of this book is to be a self-contained, practical, entry-level text on stochastic processes and control for jump diffusions in continuous time, technically Markov processes in continuous time. Sign in Help View Cart. Manage this Book. Add to my favorites. Recommend to Library.

Email to a friend. Digg This. Notify Me! E-mail Alerts. RSS Feeds. Title Information. Author s : Floyd B. Floyd B. Keywords: Stochastic processstochastic optimal controlcomputational stochastic dynamic programmingfinancial engineering applicationscomputational biomedicine applications.

Overview of This Book The aim of this book is to be a self-contained, practical, entry-level text on stochastic processes and control for jump diffusions in continuous time, technically Markov processes in continuous time.

Return to All Sections. Front Matter.Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. Bonnans Published Paris VI and Ecole Polytechnique. They give an introduction to convex analysis and its application to stochastic programming, i. This is an active subject of research that covers many applications. Save to Library. Create Alert. Launch Research Feed.

Share This Paper. References Publications referenced by this paper. Convergence of probability measures. Research Feed. SET-Valued Analysis.

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Analysis of stochastic dual dynamic programming method. Highly Influential. Convex analysis and variational problems. View 2 excerpts, references background. Integral functionals, normal integrands and measurable selections. Conjugate Duality and Optimization.

On approximate solutions of systems of linear inequalities. Related Papers. Abstract 69 References Related Papers.Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. Bertsekas Published Computer Science. This is an updated version of the research-oriented Chapter 6 on Approximate Dynamic Programming.

In addition to editorial revisions, rearrangements, and new exercises, the chapter includes an account of new research, which is collected mostly in Sections 6. Save to Library. Create Alert. Launch Research Feed. Share This Paper. Top 3 of 9 Citations View All Policy control in multiagent system with hierarchical representation.

Ariunaa Damba Zhou, W. Saad, â€¦ P. Santiago Paternain, J. Bazerque, â€¦ A. Figures and Topics from this paper. Citation Type. Has PDF. Publication Type. More Filters. Policy control in multiagent system with hierarchical representation. Research Feed. View 2 excerpts, cites background. View 1 excerpt, cites background. View 1 excerpt, cites methods.Pre-game coverage begins at 3:30 p.

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### Convex Analysis and Stochastic Programming

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