digital-brain/content/websites/data_driven_dynamical_systems_with_machine_learning.md

+++ title = "Data-Driven Dynamical Systems with Machine Learning" author = ["Dehaeze Thomas"] draft = false +++

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Data-Driven Control

Overview

Challenges

With modern control (LQR, LQG, H-Infinity), we work with linear system (or linearized systems) and we develop a control law that minimize some cost function.

Challenging systems where modern control is not efficient:

• Non-linear systems
• System with unknown dynamics
• High dimensional systems
• Limited measurements or control inputs

For these kinds of systems, data-driven control seems to be a good alternative.

What is control?

It's an optimization constrained by dynamics of the system.

The optimization is easy when the system is linear and the cost function is quadratic. When the system is non-linear or when the cost function is non quadratic, the optimization becomes complicated (non closed form). Then the optimization should be rerun on the fly, which is what is done with MPC (model predictive control).

What is Machine-Learning?

Machine-learning is powerful non-linear optimization based on data.

Outline of this lecture

Data Driven Models

For the problem of unknown dynamics, we can use data driven models. The goal is to collect data to generate of model of the system.

Machine Learning Control

When we use the control inputs, the system changes and the system model might be not valid anymore. The idea is to use data driven machine learning directly to learn a good controller.

Sensor and actuator placement

Use powerful optimization techniques from machine learning to learn what are good sensors and actuators.

Linear System Identification

The Goal of Balanced Model Reduction

Change of Variables in Control Systems

Change of Variables in Control Systems (Correction)

Balancing Example

Balancing Transformation

Balanced Truncation

Balanced Truncation Example

Error Bounds for Balanced Truncation

Balanced Proper Orthogonal Decomposition

BPOD and Output Projection

Balanced Truncation and BPOD Example

Eigensystem Realization Algorithm

ERA and the Discrete-Time Impulse Response

Eigensystem Realization Algorithm Procedure

Balanced Models with ERA

Observer Kalman Filter Identification

ERA_OKID Example in Matlab

System Identification

Full-State Models with Control

Regression Models

Dynamic Mode Decomposition with Control

DMD Control Example

Koopman with Control

Sparse Nonlinear Models with Control

Model Predictive Control

Sparse Identification of Nonlinear Dynamics for Model Predictive Control

Machine Learning Control

Overview

Genetic Algorithms

Tuning a PID Controller with Genetic Algorithms

Tuning a PID Controller with Genetic Algorithms (Part 2)

Genetic Programming

Genetic Programming Control

Extremum Seeking Control

Introduction

Matlab

Simulink

Challenging Example

Applications

Bibliography