"Stochastic Optimal Control and Reinforcement Learning", invited to speak at the Nonlinear Dynamic Systems class taught by Prof. Frank Chong-woo Park, Seoul National University, December 4, 2019.
This Presentation describes, in short, Introduction to Time Series and the overall procedure required for Time Series Modelling including general terminologies and algorithms. However the detailed Mathematics is excluded in the slides, this ppt means to give a start to understanding the Time Series Modelling before going into detailed Statistics.
This Presentation describes, in short, Introduction to Time Series and the overall procedure required for Time Series Modelling including general terminologies and algorithms. However the detailed Mathematics is excluded in the slides, this ppt means to give a start to understanding the Time Series Modelling before going into detailed Statistics.
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
International Journal of Modern Engineering Research (IJMER) covers all the fields of engineering and science: Electrical Engineering, Mechanical Engineering, Civil Engineering, Chemical Engineering, Computer Engineering, Agricultural Engineering, Aerospace Engineering, Thermodynamics, Structural Engineering, Control Engineering, Robotics, Mechatronics, Fluid Mechanics, Nanotechnology, Simulators, Web-based Learning, Remote Laboratories, Engineering Design Methods, Education Research, Students' Satisfaction and Motivation, Global Projects, and Assessment…. And many more.
Anyone who is interested in Reinforcement Learning can have a look. It covers Markov Reward Processes, Markov Decision Processes and Dynamic Programming!
A work in progress that will be abandoned as soon as this course is over. I'm not very good at P-Chem, but if you have any questions, feel free to ask. [00 CHEM 308 (1) - THIS ONE]
Maximum Entropy Reinforcement Learning (Stochastic Control)Dongmin Lee
I reviewed the following papers.
- T. Haarnoja, et al., “Reinforcement Learning with Deep Energy-Based Policies", ICML 2017
- T. Haarnoja, et al., “Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor", ICML 2018
- T. Haarnoja, et al., “Soft Actor-Critic Algorithms and Applications", arXiv preprint 2018
Thank you.
Computational Motor Control: Optimal Control for Stochastic Systems (JAIST su...hirokazutanaka
This is lecure 5 note for JAIST summer school on computational motor control (Hirokazu Tanaka & Hiroyuki Kambara). Lecture video: https://www.youtube.com/watch?v=XS7MDRMPQfU
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
International Journal of Modern Engineering Research (IJMER) covers all the fields of engineering and science: Electrical Engineering, Mechanical Engineering, Civil Engineering, Chemical Engineering, Computer Engineering, Agricultural Engineering, Aerospace Engineering, Thermodynamics, Structural Engineering, Control Engineering, Robotics, Mechatronics, Fluid Mechanics, Nanotechnology, Simulators, Web-based Learning, Remote Laboratories, Engineering Design Methods, Education Research, Students' Satisfaction and Motivation, Global Projects, and Assessment…. And many more.
Anyone who is interested in Reinforcement Learning can have a look. It covers Markov Reward Processes, Markov Decision Processes and Dynamic Programming!
A work in progress that will be abandoned as soon as this course is over. I'm not very good at P-Chem, but if you have any questions, feel free to ask. [00 CHEM 308 (1) - THIS ONE]
Maximum Entropy Reinforcement Learning (Stochastic Control)Dongmin Lee
I reviewed the following papers.
- T. Haarnoja, et al., “Reinforcement Learning with Deep Energy-Based Policies", ICML 2017
- T. Haarnoja, et al., “Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor", ICML 2018
- T. Haarnoja, et al., “Soft Actor-Critic Algorithms and Applications", arXiv preprint 2018
Thank you.
Computational Motor Control: Optimal Control for Stochastic Systems (JAIST su...hirokazutanaka
This is lecure 5 note for JAIST summer school on computational motor control (Hirokazu Tanaka & Hiroyuki Kambara). Lecture video: https://www.youtube.com/watch?v=XS7MDRMPQfU
Many Decision Problems in business and social systems can be modeled using mathematical optimization, which seeks to maximize or minimize some objective which is a function of the decisions.
Stochastic Optimization Problems are mathematical programs where some of the data incorporated into the objective or constraints are Uncertain.
whereas, Deterministic Optimization Problems are formulated with known parameters.
Kalman filter is a algorithm of predicting the future state of a system based on the previous ones.
In the presentation, I introduce to basic Kalman filtering step by step, with providing examples for better understanding.
The myphotonics project deals with the construction of opto-mechanical components and optical experiment implementation using modular systems such as LEGO®.
The components are low cost and the instructions that originated them are free to use. OpenAdaptonik and myphotonics can work together sharing the same purpose.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
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Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
15. Markov Decision Process
Markov?
“The future is independent of the past given the present”
ℙ 𝑠𝑡+1 𝑠1, … , 𝑠𝑡 = ℙ 𝑠𝑡+1 𝑠𝑡
Markov “Decision” Process:
ℙ 𝑠𝑡+1 𝑠1, 𝑎1 … , 𝑠𝑡, 𝑎 𝑡 = ℙ 𝑠𝑡+1 𝑠𝑡, 𝑎 𝑡
Future state only depends on the current state and action
&
Policy also depends on the current state only
𝜋 𝑎 𝑡 𝑠1, 𝑎1 … , 𝑠𝑡 = 𝜋 𝑎 𝑡 𝑠𝑡
19. Terminology of RL and Optimal control
State
Action
Agent
Environment
Reward of a stage
Reward (or value) function
Maximizing the value function
Bellman operator
Greedy policy w.r.t. 𝐽
State
Control Input
Controller
System
Cost of a stage
Value (or cost) function
Minimizing the value function
DP mapping or operator
Minimizing policy w.r.t. 𝐽
RL Optimal Control
27. Stochastic Optimal Control
Dynamic Programming
𝐢𝐧𝐟
𝒖 𝒌∈𝑼
𝒓 𝒙 𝒌, 𝒖 𝒌 + 𝑬 𝒙~𝒑 𝜸𝑽(𝒙 𝒌+𝟏)
How to solve the infinite-horizon discrete time system stochastic optimal control problem?
28. Stochastic Optimal Control
Dynamic Programming
𝐢𝐧𝐟
𝒖 𝒌∈𝑼
𝒓 𝒙 𝒌, 𝒖 𝒌 + 𝑬 𝒙~𝒑 𝜸𝑽(𝒙 𝒌+𝟏)
How to solve the infinite-horizon discrete time system stochastic optimal control problem?
Note) There is an other approach using different dynamic programming equation,
average reward.
Value Iteration & Policy Iteration
29. Bellman Operator
Definition. Given policy 𝜋, the state-value function 𝑉 𝜋: ℝn → ℝ is defined by
𝑉 𝜋
𝑥0 : = 𝐸 𝑥~𝑝,𝜋
𝑘=0
∞
𝛾 𝑘
𝑟𝑘(𝑥 𝑘, 𝜋(𝑥 𝑘), 𝑤 𝑘) ȁ 𝑥 = 𝑥0 at 𝑡 = 0
= 𝑟(𝑥0, 𝜋(𝑥 𝑘)) + 𝐸 𝑥~𝑝,𝜋
𝑘=0
∞
𝛾 𝑘
𝑟𝑘(𝑥 𝑘, 𝜋(𝑥 𝑘), 𝑤 𝑘)
and the state-input-value function 𝑄 𝜋
: ℝn
× ℝm
→ ℝ is defined by
𝑄 𝜋 𝑥0, 𝑢0 : = 𝐸 𝑥~𝑝,𝜋
𝑘=0
∞
𝛾 𝑘 𝑟𝑘(𝑥 𝑘, 𝜋(𝑥 𝑘), 𝑤 𝑘) ȁ 𝑥 = 𝑥0, 𝑢 = 𝑢0 at 𝑡 = 0
= 𝑟 𝑥0, 𝑢0 + 𝐸 𝑥~𝑝,𝜋
𝑘=1
∞
𝛾 𝑘 𝑟𝑘(𝑥 𝑘, 𝜋(𝑥 𝑘), 𝑤 𝑘)
34. Bellman Operator
Proposition 1. (Monotonicity) The Bellman operator 𝑇 𝜋
, 𝑇∗
are monotone, i.e. if
𝜓 𝑥 ≤ 𝜓′ 𝑥 ∀𝑥 ∈ 𝑋,
then
𝑇 𝜋
𝜓 𝑥 ≤ 𝑇 𝜋
𝜓′ 𝑥 ∀𝑥 ∈ 𝑋
𝑇∗
𝜓 𝑥 ≤ 𝑇∗
𝜓′ 𝑥 ∀𝑥 ∈ 𝑋.
Proposition 2. (Constant shift property) For any scalar 𝑟,
𝑇 𝜋
𝜓 + 𝑟𝑒 𝑥 = 𝑇 𝜋
𝜓 𝑥 + 𝛾𝑟 ∀𝑥 ∈ 𝑋
𝑇∗
𝜓 + 𝑟𝑒 𝑥 = 𝑇∗
𝜓 𝑥 + 𝛾𝑟 ∀𝑥 ∈ 𝑋.
Proposition 3. The Bellman operator 𝑇 𝜋
, 𝑇 are a contraction
with modulus 𝛾 with respect to the sup norm ∙ ∞ i.e.
𝑇 𝜋 𝜓 − 𝑇 𝜋 𝜓′ ≤ 𝛾 𝜓 − 𝜓′ 𝜓, 𝜓′ ∈ 𝔹
𝑇∗ 𝜓 − 𝑇∗ 𝜓′ ≤ 𝛾 𝜓 − 𝜓′ 𝜓, 𝜓′ ∈ 𝔹.
35. Bellman Operator
Theorem 2.3. (Contraction Mapping Theorem) Let 𝔹, ∙ ∞, 𝑑∞ be a metric
space and 𝑇: 𝔹 → 𝔹 is a contraction mapping with modulus 𝛾. Then,
1) 𝑇 has an unique fixed point in 𝔹 i.e. there exist unique 𝑓∗
∈ 𝔹 s.t. 𝑇𝑓∗
= 𝑓∗
.
2) Consider the sequence {𝑓𝑛} s.t. 𝑓𝑛+1 = 𝑇𝑓𝑛 ∀𝑓0 ∈ 𝔹.
Then, lim
n→∞
𝑇 𝑛
𝑓0 → 𝑓∗
.
36. Value Iteration
Algorithm: Value Iteration
Input: 𝑟, 𝑝, 𝛾, 𝛥
Output: 𝜋
1. 𝑉 𝑥 ←Initialize arbitrarily for 𝑥 ∈ 𝑋
2. repeat
3. for all 𝑥 ∈ 𝑋 do
4. 𝑉𝑘+1 ← 𝑇∗
𝑉𝑘
5. until 𝑉𝑘+1 − 𝑉𝑘 < 𝛥
6. 𝜋∗ 𝑥 ∈ 𝑎𝑟𝑔min
𝑢∈𝑈(𝑥)
𝑟 𝑥, 𝑢 + 𝛾 σ 𝑥′∈𝑋 𝑝 𝑥′ 𝑥, 𝑢 𝑉𝑘(𝑥′)
7. return 𝜋∗
41. Learning-based approach
Q: If the system dynamics 𝑝 and reward function 𝑟 is unknown,
how do we solve DP equation?
1. Estimate model (𝑟 and 𝑝) from simulation data and use previous methods
Model-based approach (model learning)
42. Learning-based approach
Q: If the system dynamics 𝑝 and reward function 𝑟 is unknown,
how do we solve DP equation?
1. Estimate model (𝑟 and 𝑝) from simulation data and use previous methods
Model-based approach (model learning)
2. Without system identification, obtain the value function and policy directly from simulation data
Model-free approach
44. 𝐢𝐧𝐟
𝝅∈𝚷
𝑬 𝝅 𝒓 𝒙 𝒌, 𝝅(𝒙 𝒌) + 𝑬 𝒙~𝒑 𝜸𝑽(𝒙 𝒌+𝟏)
Approximation in policy space
Approximation in value space
Approximate 𝐸[∙]
Parametric approximation
Problem approximation
Rollout, MPC
Monte-Carlo search
Certainty equivalence
Policy search
Policy gradient
45. Approximation
in Value space
Approximation
in Policy space
Actor-Critic
• TD, SARSA, Q-learning
• Function approximation
• Policy search
• Policy gradient
DPG, DDPG, TRPO, CPO, PPO, Soft actor-critic, …
Approximate
Expectation
• Monte-Carlo search
• Certainty equivalence
46. Approximation in Value Space
DP algorithms sweep over “all state” for each step
𝐸 𝑓 ≈
1
𝑁
𝑖=1
𝑁
𝑓𝑖Use Monte-Carlo Search
47. Approximation in Value Space
DP algorithms sweep over “all state” for each step
𝑁~14,000,605
Use Monte-Carlo Search 𝐸 𝑓 ≈
1
𝑁
𝑖=1
𝑁
𝑓𝑖
48. Approximation in Value Space
DP algorithms sweep over “all state” for each step
𝑁~14,000,605
Use Monte-Carlo Search 𝐸 𝑓 ≈
1
𝑁
𝑖=1
𝑁
𝑓𝑖
Impractical
49. Stochastic Approximation
Consider the problem
𝑥 = 𝐿(𝑥).
Then, this problem can be solved by iterative algorithm
𝑥 𝑘+1 = 𝐿 𝑥 𝑘
or
𝑥 𝑘+1 = 1 − 𝛼 𝑘 𝑥 𝑘 + 𝛼 𝑘 𝐿(𝑥 𝑘).
If 𝐿 𝑥 is of the form 𝐸 𝑓(𝑥, 𝑤) where 𝑤 is a random noise. Then, 𝐿(𝑥) can be approximated by
𝐿 𝑥 =
1
𝑁
𝑖=1
𝑁
𝑓(𝑥𝑖, 𝑤𝑖)
which becomes inefficient when 𝑁 is large.
50. Stochastic Approximation
Use a single sample as an estimation of expectation in each update
This update can be seen as a stochastic approximation of the form
𝑥 𝑘+1 = 1 − 𝛼 𝑘 𝑥 𝑘 + 𝛼 𝑘 𝑓(𝑥 𝑘, 𝑤 𝑘).
𝑥 𝑘+1 = 1 − 𝛼 𝑘 𝑥 𝑘 + 𝛼 𝑘 𝐸 𝑓 𝑥 𝑘, 𝑤 𝑘 + 𝜀 𝑘
= 1 − 𝛼 𝑘 𝑥 𝑘 + 𝛼 𝑘 𝐿(𝑥 𝑘) + 𝜀 𝑘
where 𝜀 𝑘= 𝑓 𝑥 𝑘, 𝑤 𝑘 − 𝐸 𝑓 𝑥 𝑘, 𝑤 𝑘 .
Robbins-Monro stochastic approximation guarantees the convergence under
contraction or monotonicity assumptions of the mapping 𝐿 with assumptions
𝑘=0
∞
𝛼 𝑘 = + ∞ and
𝑘=0
∞
𝛼 𝑘
2
< +∞.