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LEARNING IN ADAPTIVE SYSTEMS
2.153 Lecture 5, Fall 2019 1
Adaptive Control
Parameter estimate
Model Structure
On‐line information
Control input The design of a self‐tuning controller in the presence of parametric uncertainties
Learning in Adaptive Systems
Adaptive Estimation
Parameter estimate
Model Structure
On‐line information
State estimate Generate real‐time estimates of parameters and states using on‐line data
2.153 Lecture 5, Fall 2019 2
Learning ≡Parameter Estimation in Real‐time
: parameter, unknown
, , : state, output, input
Problem Statement – Adaptive Estimation
Goal: Find , so that → , → Γ
: state error
: parameter estimate
Adaptive Estimator Parameter estimate
Model Structure
On‐line information
State estimate
Plant(Dynamic System)
, ,
2.153 Lecture 5, Fall 2019 3
: parameter, unknown
, , : state, output, input
: real‐time data
Problem Statement – Adaptive Control
Goal: Find , , so that regulation and tracking occur
: tracking error
c: control parameter estimate
Adaptive Control Parameter estimate
Model Structure
On‐line information
Control input
Plant(Dynamic System)
2.153 Lecture 5, Fall 2019
+
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Error Models ‐ Two types of errors
• : Performance error (ex. ; o can be measured, needs to be reduced
• : Parameter error (ex. o Unknown, can be adjusted – Learning Rule
Goal: Determine error models and learning rules 2.153 Lecture 5, Fall 2019
Error ModelPerformance errorReal‐time data
Adaptive Estimation
Parameter estimate
Model Structure
On-line information
State estimate
Adaptive Control
Parameter estimate
Model Structure
On-line information
Control input
5
Typical Error Models
Goal: Find learning rules for adjusting so that high performance and learning are achieved
: tracking error, ∗
Adaptive Control Parameter estimate
Model Structure
On‐line information
Control input
Plant(Dynamic System)
Error Model 1: : system data
Error Model 3: : Dynamic operator
2.153 Lecture 5, Fall 2019 6
Error Model 1
2.153 Lecture 5, Fall 2019
2 2 2 07
Build an estimator: Parameter error:
Performance error:
Error Model:
Simplest Gradient Descent
Stability Framework:
System Model:
• Can be extended to higher order systems (in a later lecture)• Class of strictly positive real
⇒ and have same sign most of the time
• , But the adaptive law still works
• If is not strictly positive real, this will not work
Error Model 3
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′
0
1
⇒ 12
Stability Framework
Goal: Find learning rules for adjusting so that
• Performance and Learning are conflicting objectives
• Effect of imperfect learning on performance has to be accommodated.
• Use a stability framework
2.153 Lecture 5, Fall 2019 9
Performance errorReal‐time data
Distinctions: Gradient‐descent leads to instability!
2.153 Lecture 5, Fall 2019
2 2∗→Instability
2 0(for a class of )
Stable adaptive law
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Performance errorReal‐time data
* HP Whitaker, An Adaptive System for Control of the Dynamics Performances of Aircraft and Spacecraft, Inst Aeronautical Services, 1959 ‐ MIT Rule
Guarantees with Imperfect Learning
2.153 Lecture 5, Fall 2019
Stable Subspace Unstable Subspace∗
Adaptive Control
∆
Adaptive,
With
Open Loop Unstable Short Period Dynamics
ControllerLearningdata
∆
Performance goal:Adaptive Controller
Imperfect learning, yet safe performance
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MACHINE LEARNING
2.153 Lecture 5, Fall 2019 12
Machine LearningThe ability of a computer to learn using on‐line data• Significant success in image & speech recognition, games• By and large a classification/pattern‐recognition tool
Standard Methods• Linear Regression• Logistic Regression• Multi‐layered Neural Networks• Support‐Vector Machines
Typical learning methods:• Supervised learning• Unsupervised learning• Reinforcement learning
2.153 Lecture 5, Fall 2019 13
Neural Networks: A popular learning methodology
A bit of history• Proposed in 1944 by McCullough and Pitts• Controversy in the ‘70s: Multilayered Perceptrons, Minsky and Papert
• Resurgence in the 1980s• A re‐resurgence in the 21st century – fast processing power of graphics
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(from MIT News)
Fundamentals of ML/Neural Networks
Universal Approximation Theorem
(From YouTube, Why do Nnets work?)
Neural Network
∗ ∗
∗(One hidden layer) . : activation function
2.153 Lecture 5, Fall 2019 15
Activation functions
2.153 Lecture 5, Fall 2019
(from Wikipedia)
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Typical Examples
(1) : demographic/employment related variable: income above or below $50
(2) : 15,000 nouns, 500 images from the web: ranking based on # occurrences
(3) : 800,000 text articles: classify into 4 categories
(4) : pixel images: classify into one of 10 digits 0,… , 9
2.153 Lecture 5, Fall 2019 17
Deep Learning: Increased layers
2.153 Lecture 5, Fall 2019 18
(adapted from P. Khargonekar, ACC Workshop on Machine Learning, 2018)