Optimal and Autonomous Control Using Reinforcement Learning: a Survey
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Propt Product Sheet
PROPT - The world’s fastest Optimal Control platform for MATLAB. PROPT - ONE OF A KIND, LIGHTNING FAST SOLUTIONS TO YOUR OPTIMAL CONTROL PROBLEMS! NOW WITH WELL OVER 100 TEST CASES! The PROPT software package is intended to When using PROPT, optimally coded analytical solve dynamic optimization problems. Such first and second order derivatives, including problems are usually described by: problem sparsity patterns are automatically generated, thereby making it the first MATLAB • A state-space model of package to be able to fully utilize a system. This can be NLP (and QP) solvers such as: either a set of ordinary KNITRO, CONOPT, SNOPT and differential equations CPLEX. (ODE) or differential PROPT currently uses Gauss or algebraic equations (PAE). Chebyshev-point collocation • Initial and/or final for solving optimal control conditions (sometimes problems. However, the code is also conditions at other written in a more general way, points). allowing for a DAE rather than an ODE formulation. Parameter • A cost functional, i.e. a estimation problems are also scalar value that depends possible to solve. on the state trajectories and the control function. PROPT has three main functions: • Sometimes, additional equations and variables • Computation of the that, for example, relate constant matrices used for the the initial and final differentiation and integration conditions to each other. of the polynomials used to approximate the solution to the trajectory optimization problem. The goal of PROPT is to make it possible to input such problem • Source transformation to turn descriptions as simply as user-supplied expressions into possible, without having to worry optimized MATLAB code for the about the mathematics of the cost function f and constraint actual solver. -
Calculating the Value Function When the Bellman Equation Cannot Be
Journal of Economic Dynamics & Control 28 (2004) 1437–1460 www.elsevier.com/locate/econbase Investment under uncertainty: calculating the value function when the Bellman equation cannot besolvedanalytically Thomas Dangla, Franz Wirlb;∗ aVienna University of Technology, Theresianumgasse 27, A-1040 Vienna, Austria bUniversity of Vienna, Brunnerstr.˝ 72, A-1210 Vienna, Austria Abstract The treatment of real investments parallel to ÿnancial options if an investment is irreversible and facing uncertainty is a major insight of modern investment theory well expounded in the book Investment under Uncertainty by Dixit and Pindyck. Thepurposeof this paperis to draw attention to the fact that many problems in managerial decision making imply Bellman equations that cannot be solved analytically and to the di6culties that arise when approximating therequiredvaluefunction numerically.This is so becausethevaluefunctionis thesaddlepoint path from the entire family of solution curves satisfying the di7erential equation. The paper uses a simple machine replacement and maintenance framework to highlight these di6culties (for shooting as well as for ÿnite di7erence methods). On the constructive side, the paper suggests and tests a properly modiÿed projection algorithm extending the proposal in Judd (J. Econ. Theory 58 (1992) 410). This fast algorithm proves amendable to economic and stochastic control problems(which onecannot say of theÿnitedi7erencemethod). ? 2003 Elsevier B.V. All rights reserved. JEL classiÿcation: D81; C61; E22 Keywords: Stochastic optimal control; Investment under uncertainty; Projection method; Maintenance 1. Introduction Without doubt, thebook of Dixit and Pindyck (1994) is oneof themost stimulating books concerning management sciences. The basic message of this excellent book is the following: Considering investments characterized by uncertainty (either with ∗ Corresponding author. -
Optimal Control Theory Version 0.2
An Introduction to Mathematical 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 Chapter 7: Introduction to stochastic control theory Appendix: Proofs of the Pontryagin Maximum Principle Exercises References 1 PREFACE These notes build upon a course I taught at the University of Maryland during the fall of 1983. My great thanks go to Martino Bardi, who took careful notes, saved them all these years and recently mailed them to me. Faye Yeager typed up his notes into a first draft of these lectures as they now appear. Scott Armstrong read over the notes and suggested many improvements: thanks, Scott. Stephen Moye of the American Math Society helped me a lot with AMSTeX versus LaTeX issues. My thanks also to Atilla Yilmaz for spotting lots of typos and errors, which I have corrected. I have radically modified much of the notation (to be consistent with my other writings), updated the references, added several new examples, and provided a proof of the Pontryagin Maximum Principle. As this is a course for undergraduates, I have dispensed in certain proofs with various measurability and continuity issues, and as compensation have added various critiques as to the lack of total rigor. This current version of the notes is not yet complete, but meets I think the usual high standards for material posted on the internet. Please email me at [email protected] with any corrections or comments. -
From Single-Agent to Multi-Agent Reinforcement Learning: Foundational Concepts and Methods Learning Theory Course
From Single-Agent to Multi-Agent Reinforcement Learning: Foundational Concepts and Methods Learning Theory Course Gon¸caloNeto Instituto de Sistemas e Rob´otica Instituto Superior T´ecnico May 2005 Abstract Interest in robotic and software agents has increased a lot in the last decades. They allow us to do tasks that we would hardly accomplish otherwise. Par- ticularly, multi-agent systems motivate distributed solutions that can be cheaper and more efficient than centralized single-agent ones. In this context, reinforcement learning provides a way for agents to com- pute optimal ways of performing the required tasks, with just a small in- struction indicating if the task was or was not accomplished. Learning in multi-agent systems, however, poses the problem of non- stationarity due to interactions with other agents. In fact, the RL methods for the single agent domain assume stationarity of the environment and cannot be applied directly. This work is divided in two main parts. In the first one, the reinforcement learning framework for single-agent domains is analyzed and some classical solutions presented, based on Markov decision processes. In the second part, the multi-agent domain is analyzed, borrowing tools from game theory, namely stochastic games, and the most significant work on learning optimal decisions for this type of systems is presented. ii Contents Abstract 1 1 Introduction 2 2 Single-Agent Framework 5 2.1 Markov Decision Processes . 6 2.2 Dynamic Programming . 10 2.2.1 Value Iteration . 11 2.2.2 Policy Iteration . 12 2.2.3 Generalized Policy Iteration . 13 2.3 Learning with Model-free methods . -
Dynamic Programming for Dummies, Parts I & II
DYNAMIC PROGRAMMING FOR DUMMIES Parts I & II Gonçalo L. Fonseca [email protected] Contents: Part I (1) Some Basic Intuition in Finite Horizons (a) Optimal Control vs. Dynamic Programming (b) The Finite Case: Value Functions and the Euler Equation (c) The Recursive Solution (i) Example No.1 - Consumption-Savings Decisions (ii) Example No.2 - Investment with Adjustment Costs (iii) Example No. 3 - Habit Formation (2) The Infinite Case: Bellman's Equation (a) Some Basic Intuition (b) Why does Bellman's Equation Exist? (c) Euler Once Again (d) Setting up the Bellman's: Some Examples (i) Labor Supply/Household Production model (ii) Investment with Adjustment Costs (3) Solving Bellman's Equation for Policy Functions (a) Guess and Verify Method: The Idea (i) Guessing the Policy Function u = h(x) (ii) Guessing the Value Function V(x). (b) Guess and Verify Method: Two Examples (i) One Example: Log Return and Cobb-Douglas Transition (ii) The Other Example: Quadratic Return and Linear Transition Part II (4) Stochastic Dynamic Programming (a) Some Basics (b) Some Examples: (i) Consumption with Uncertainty (ii) Asset Prices (iii) Search Models (5) Mathematical Appendices (a) Principle of Optimality (b) Existence of Bellman's Equation 2 Texts There are actually not many books on dynamic programming methods in economics. The following are standard references: Stokey, N.L. and Lucas, R.E. (1989) Recursive Methods in Economic Dynamics. (Harvard University Press) Sargent, T.J. (1987) Dynamic Macroeconomic Theory (Harvard University Press) Sargent, T.J. (1997) Recursive Macroeconomic Theory (unpublished, but on Sargent's website at http://riffle.stanford.edu) Stokey and Lucas does more the formal "mathy" part of it, with few worked-through applications. -
The Uncertainty Bellman Equation and Exploration
The Uncertainty Bellman Equation and Exploration Brendan O’Donoghue 1 Ian Osband 1 Remi Munos 1 Volodymyr Mnih 1 Abstract tions that maximize rewards given its current knowledge? We consider the exploration/exploitation prob- Separating estimation and control in RL via ‘greedy’ algo- lem in reinforcement learning. For exploitation, rithms can lead to premature and suboptimal exploitation. it is well known that the Bellman equation con- To offset this, the majority of practical implementations in- nects the value at any time-step to the expected troduce some random noise or dithering into their action value at subsequent time-steps. In this paper we selection (such as -greedy). These algorithms will even- consider a similar uncertainty Bellman equation tually explore every reachable state and action infinitely (UBE), which connects the uncertainty at any often, but can take exponentially long to learn the opti- time-step to the expected uncertainties at subse- mal policy (Kakade, 2003). By contrast, for any set of quent time-steps, thereby extending the potential prior beliefs the optimal exploration policy can be com- exploratory benefit of a policy beyond individual puted directly by dynamic programming in the Bayesian time-steps. We prove that the unique fixed point belief space. However, this approach can be computation- of the UBE yields an upper bound on the vari- ally intractable for even very small problems (Guez et al., ance of the posterior distribution of the Q-values 2012) while direct computational approximations can fail induced by any policy. This bound can be much spectacularly badly (Munos, 2014). tighter than traditional count-based bonuses that For this reason, most provably-efficient approaches to re- compound standard deviation rather than vari- inforcement learning rely upon the optimism in the face of ance. -
Macroeconomics: a Dynamic General Equilibrium Approach
Macroeconomics: A Dynamic General Equilibrium Approach Mausumi Das Lecture Notes, DSE Jan 29-Feb 19, 2019 Das (Lecture Notes, DSE) DGE Approach Jan29-Feb19,2019 1/113 Modern Macroeconomics: the Dynamic General Equilibrium (DGE) Approach Modern macroeconomics is based on a dynamic general equilibrium approach which postulates that Economic agents are continuously optimizing/re-optimizing subject to their constraints and subject to their information set. They optimize not only over their current choice variables but also the choices that would be realized in future. All agents have rational expectations: thus their ex ante optimal future choices would ex post turn out to be less than optimal if and only if their information set is incomplete and/or there are some random elements in the economy which cannot be anticipated perfectly. The agents are atomistic is the sense that they treat the market factors as exogenous in their optimization exercise. The optimal choices of all agents are then mediated through the markets to produce an equilibrium outcome for the macroeconomy (which, by construction, is also consistent with the optimal choice of each agent). Das (Lecture Notes, DSE) DGE Approach Jan29-Feb19,2019 2/113 Modern Macroeconomics: DGE Approach (Contd.) This approach is ‘dynamic’because agents are making choices over variables that relate to both present and future. This approach is ‘equilibrium’because the outcome for the macro-economy is the aggregation of individuals’equilibrium (optimal) behaviour. This approach is ‘general equilibrium’because it simultaneously takes into account the optimal behaviour of diiferent types of agents in different markets and ensures that all markets clear. -
Economics 2010C: Lecture 1 Introduction to Dynamic Programming
Economics 2010c: Lecture 1 Introduction to Dynamic Programming David Laibson 9/02/2014 Outline of my half-semester course: 1. Discrete time methods (Bellman Equation, Contraction Mapping Theorem, and Blackwell’s Sufficient Conditions, Numerical methods) Applications to growth, search, consumption, asset pricing • 2. Continuous time methods (Bellman Equation, Brownian Motion, Ito Process, and Ito’s Lemma) Applications to search, consumption, price-setting, investment, indus- • trial organization, asset-pricing Outline of today’s lecture: 1. Introduction to dynamic programming 2. The Bellman Equation 3. Three ways to solve the Bellman Equation 4. Application: Search and stopping problem 1 Introduction to dynamic programming. Course emphasizes methodological techniques and illustrates them through • applications. We start with discrete-time dynamic optimization. • Is optimization a ridiculous model of human behavior? Why or why not? • Today we’ll start with an -horizon stationary problem: • ∞ The Sequence Problem (cf. Stokey and Lucas) Notation: is the state vector at date (+1) is the flow payoff at date ( is ‘stationary’) is the exponential discount function is referred to as the exponential discount factor The discount rate is the rate of decline of the discount function, so ln = ≡− − h i Note that exp( )= and exp( )= − − Definition of Sequence Problem: Find () such that ∞ (0)= sup (+1) +1 =0 { }∞=0 X subject to +1 Γ() with 0 given. ∈ Remark 1.1 When I omit time subscripts, this implies that an equation holds for all relevant values of . In the statement above, +1 Γ() implies, ∈ +1 Γ() for all =0 1 2 ∈ Example 1.1 Optimal growth with log utility and Cobb-Douglas technology: ∞ sup ln() =0 { }∞=0 X subject to the constraints, 0+ +1 = and 0 given. -
Graduate Macro Theory II: Notes on Investment
Graduate Macro Theory II: Notes on Investment Eric Sims University of Notre Dame Spring 2011 1 Introduction These notes introduce and discuss modern theories of firm investment. While much of this is done as a decision rule problem of the firm, it is easily incorporated into a general equilibrium structure. 2 Tobin's Q Jim Tobin (1969) developed an intuitive and celebrated theory of investment. He reasoned that if the market value of physical capital of a firm exceeded its replacement cost, then capital has more value \in the firm” (the numerator) than outside the firm (the denominator). Formally, Tobin's Q is defined as: Market Value of Firm Capital Q = (1) Replacement Cost of Capital Tobin reasoned that firms should accumulate more capital when Q > 1 and should draw down their capital stock when Q < 1. That is, net investment in physical capital should depend on where Q is in relation to one. How would one measure Q in the data? Typically this is done by using the value of the stock market in the numerator and data on corporate net worth in the denominator { here net worth is defined as the replacement cost of capital goods. For example, suppose that a company is in the business of bulldozing. It owns a bunch of bulldozers. Suppose that it owns 100 bulldozers, which cost 100 each to replace. This means its replacement cost is 10; 000. Suppose it has 1000 outstanding shares of stock, valued at 15 per share. The market value of the firm is thus 15; 000 and the replacement cost is 10; 000, so Tobin's Q is 1.5. -
Ensemble Kalman Filtering for Inverse Optimal Control Andrea Arnold and Hien Tran, Member, SIAM
Ensemble Kalman Filtering for Inverse Optimal Control Andrea Arnold and Hien Tran, Member, SIAM Abstract—Solving the inverse optimal control problem for In the Bayesian framework, unknown parameters are mod- discrete-time nonlinear systems requires the construction of a eled as random variables with probability density functions stabilizing feedback control law based on a control Lyapunov representing distributions of possible values. The EnKF is a function (CLF). However, there are few systematic approaches available for defining appropriate CLFs. We propose an ap- nonlinear Bayesian filter which uses ensemble statistics in proach that employs Bayesian filtering methodology to param- combination with the classical Kalman filter equations for eterize a quadratic CLF. In particular, we use the ensemble state and parameter estimation [9]–[11]. The EnKF has been Kalman filter (EnKF) to estimate parameters used in defining employed in many settings, including weather prediction the CLF within the control loop of the inverse optimal control [12], [13] and mathematical biology [11]. To the authors’ problem formulation. Using the EnKF in this setting provides a natural link between uncertainty quantification and optimal knowledge, this is the first proposed use of the EnKF in design and control, as well as a novel and intuitive way to find inverse optimal control problems. The novelty of using the the one control out of an ensemble that stabilizes the system the EnKF in this setting allows us to generate an ensemble of fastest. Results are demonstrated on both a linear and nonlinear control laws, from which we can then select the control law test problem. that drives the system to zero the fastest. -
Optimal Control Theory
Optimal Control Theory Mrinal Kanti Ghosh Department of Mathematics Indian Institute of Science Bangalore 560 012 email : [email protected] 1 Introduction Optimal Control Theory deals with optimization problems involving a controlled dynamical system. A controlled dynamical system is a dynamical system in which the trajectory can be altered continuously in time by choosing a control parameter u(t) continuously in time. A (deterministic) controlled dynamical system is usually governed by an ordinary differential equation of the form 9 x_(t) = f(t; x(t); u(t)); t > 0 = (1:1) d ; x(0) = x0 2 R where u : Rd ! U is a function called the control, U a given set called the control set, and d d f : R+ × R × U ! IR : By choosing the value of u(t), the state trajectory x(t) can be controlled. The objective of controlling the state trajectory is to minimize a certain cost associated with (1.1). Consider the following important problem studied by Bushaw and Lasalle in the field of aerospace engineering. Let x(t) represent the deviation of a rocket trajectory from some desired flight path. Then, provided the deviation is small, it is governed to a good approxi- mation by 9 x_(t) = A(t)x(t) + B(t)u(t); t > 0 = x(0) = x0; x(T ) = 0; ; (1:2) u(t) 2 [−1; 1]; where A and B are appropriate matrix valued functions. The vector x0 is the initial deviation, u(t) is the rocket thrust at time t, and T is the final time. -
PSO-Based Soft Lunar Landing with Hazard Avoidance: Analysis and Experimentation
aerospace Article PSO-Based Soft Lunar Landing with Hazard Avoidance: Analysis and Experimentation Andrea D’Ambrosio 1,* , Andrea Carbone 1 , Dario Spiller 2 and Fabio Curti 1 1 School of Aerospace Engineering, Sapienza University of Rome, Via Salaria 851, 00138 Rome, Italy; [email protected] (A.C.); [email protected] (F.C.) 2 Italian Space Agency, Via del Politecnico snc, 00133 Rome, Italy; [email protected] * Correspondence: [email protected] Abstract: The problem of real-time optimal guidance is extremely important for successful au- tonomous missions. In this paper, the last phases of autonomous lunar landing trajectories are addressed. The proposed guidance is based on the Particle Swarm Optimization, and the differ- ential flatness approach, which is a subclass of the inverse dynamics technique. The trajectory is approximated by polynomials and the control policy is obtained in an analytical closed form solution, where boundary and dynamical constraints are a priori satisfied. Although this procedure leads to sub-optimal solutions, it results in beng fast and thus potentially suitable to be used for real-time purposes. Moreover, the presence of craters on the lunar terrain is considered; therefore, hazard detection and avoidance are also carried out. The proposed guidance is tested by Monte Carlo simulations to evaluate its performances and a robust procedure, made up of safe additional maneuvers, is introduced to counteract optimization failures and achieve soft landing. Finally, the whole procedure is tested through an experimental facility, consisting of a robotic manipulator, equipped with a camera, and a simulated lunar terrain. The results show the efficiency and reliability Citation: D’Ambrosio, A.; Carbone, of the proposed guidance and its possible use for real-time sub-optimal trajectory generation within A.; Spiller, D.; Curti, F.