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Session 6 Least squares problems. Adjoint operator. Reading Assignment Luenberger “Optimization by Vector Space Methods” Chapter 6. The link for the book PDF: https://www.dropbox.com/s/82czmjit8ic4oay/David%20G.%20Luenberger% 20-%20Optimization%20by%20Vector%20Space%20Methods.pdf?dl=0 Exercise 6.1 [Reading] Least squares solution to linear equations also has a clear interpretation by using project
LionSealWhite Linear Systems Lecturer: Zhiyong Sun Department Automatic Control Lund University 1 / 25 LionSealWhite Lecture 0 Course contents Some math background Vector spaces and mappings Matrix theory Norms Material: Lecture slides R.A. Horn and C.R. Johnson. Matrix Analysis. Cambridge University Press, 2013. 2 / 25 LionSealWhite Linear Systems, 2019 Introduction Multivariable Time-varying Sys
LionSealWhite Linear Systems, 2019 - Lecture 2 Transition Matrix Properties Time-varying change of coordinates Periodic Systems Floquet Decomposition Time-varying Transfer Functions Rugh, Chapter 5 [and Chapter 21] Main news: Properties of LTV systems LTP systems 1 / 41 LionSealWhite Continuous Time-varying (LTV) Systems For bounded A(t), the equation ẋ(t) = A(t)x(t), x(t0) = x0 has a unique solu
LionSealWhite Linear Systems, 2019 - Lecture 4 Realization from Weighting Pattern Minimal Realizations Realization from Transfer Function Realization from Markov Parameters Discrete Time Rugh Ch 10, 11 (only pp194-199, skip proof of 11.7), (26) 1 / 30 LionSealWhite Example: Shift Register Synthesis x1 x2 x3 x4 x = [ x1 x2 x3 x4 ]T x(k + 1) = 1 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 x(k) +
LionSealWhite Lecture 6 Least squares problems Adjoint operators 1 / 32 LionSealWhite Review: Least Squares Solution to Linear Equations (I) Consider a system of linear equations Ax = b, A ∈ Rm×n, b ∈ Rm with m ≥ n and rank(A) = n (Tall A—more rows than columns, or more equations than unknowns). If b /∈ range(A) then the linear system is inconsistent, i.e., no solution exists. Find x that minimize
LionSealWhite Lecture 8 Differential Algebraic Equations Rosenbrock System Matrix Course Review Suggested reading: T. Kailath Linear Systems, Chapter 8 (link available in the email). 1 / 26 LionSealWhite Differential Algebraic Equation Models of physical systems are often on the form 0 = F (ẋ, x, t) If x and ẋ enter linearly we get Eẋ = Ax+ f(t) Linear Differential Algebraic Equation (DAE) Any
6 LINEAR OPERATORS AND ADJOINTS 6.1 Introduction A study of linear operators and adjoints is essential for a sophisticated approach to many problems oflinear vector spaces. The associated concepts and notations of operator theory often streamline an otherwise cumber some analysis by eliminating the need for carrying along complicated explicit formulas and by enhancing one's insight of the problem
Adaptive Control K. J. Åström Department of Automatic Control, LTH Lund University October 23, 2020 Adaptive Control 1. Introduction 2. Self-oscillating Adaptive Control 3. Model Reference Adaptive Control 4. Estimation and Excitation 5. Minimum Variance Control 6. Self-Tuning Regulators 7. Learning and Dual Control 8. Applications 9. Related Fields 10. Summary Introduction Adapt to adjust to a sp
Control System Synthesis - Basics - PhD Class - Fall 2020 Control System Synthesis - Basics PHD CLASS - FALL 2020 Representation of feedback systems Block diagrams System description Feedback fundamentals Static analysis Design issues The Gang of Four Stability and Nyquist plots Performances and Bode plots About feedforward Conclusions 1 Introduction 2 Fundamentals: problem formulation System repr
Control System Synthesis - Optimal control and LQG - PhD Class - Fall 2020 Control System Synthesis - Optimal control and LQG PHD CLASS - FALL 2020 The optimal control problem LQG control What is LQG control? Controllability and LQR Observability and state estimation Summary Optimal control Dynamic programming and HJB Indirect methods and Pontryagin’s principle Summary 1 Introduction 2 Fundamental
Control System Synthesis - Robust control - PhD Class - Fall 2020 Control System Synthesis - Robust control PHD CLASS - FALL 2020 Uncertainty and robustness Where does uncertainty come from? Modelling uncertainty Robustness Small gain theorem Robust stability Robust performance Robust synthesis H∞ -synthesis H∞ -Loopshaping synthesis µ-analysis and synthesis 1 Introduction 2 Fundamentals 3 Design
Control System Synthesis - Data-driven control - PhD Class - Fall 2020 Control System Synthesis - Data-driven control PHD CLASS - FALL 2020 Introduction to data-driven control The importance of data-driven approaches Model-based and data-driven control Overview of data-driven control technique Predictive and learning DDC Use of local models Use of repetitive experiments Robust DDC Using convex opt
PID Control Karl Johan Åström Tore Hägglund Department of Automatic Control, Lund University September 23, 2020 PID Control 1. Introduction 2. The Controller 3. Stability 4. Performance and Robustness 5. Empirical Tuning Rules 6. Tuning based on Optimization 7. Relay Auto-tuning 8. Limitations of PID Control 9. Summary Theme: The most common controller. Introduction ◮ PID control is widely used in
https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/ControlSystemsSynthesis/2020/PIDeight.pdf - 2025-05-04
Control System Synthesis - PhD Class Exercise session 2 October 8, 2020 1 Inverted pendulum on a cart Figure 1: Inverted pendulum. The equations of motion are : (M +m)ẍ+ bẋ+mlθ̈ cos θ −mlθ̇2 sin θ = F (J +ml2)θ̈ +mgl sin θ = −mlẍ cos θ (1) where: • M = 0.5kg is the mass of the cart • m = 0.2kg is the mass of the pendulum • b = 0.1N/m/sec is the coefficient of friction for the cart • l = 0.3m is
Control System Synthesis - PhD Class Handin 1: Temperature control in a heat exchanger 24/09/2020 A chemical reactor called “stirring tank” is depicted below. The top inlet delivers liquid to be mixed in the tank. The tank liquid must be maintained at a constant temperature by varying the amount of steam supplied to the heat exchanger (bottom pipe) via its control valve. Variations in the temperat
flexservo.dvi Handin - Flexible Servo The process consists of three horizontal pulleys connected by two elastic belts. SensorDC motor The transfer function from motor to sensor can take 3 forms (Ts = 50ms): Unloaded: B = 0.28261z−3 + 0.50666z−4 A = 1− 1.41833z−1 + 1.58939z−2 − 1.31608z−3 + 0.88642z−4 Half Load: B = 0.10276z−3 + 0.18123z−4 A = 1− 1.99185z−1 + 2.20265z−2 − 1.84083z−3 + 0.89413z−4 Fu
https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/ControlSystemsSynthesis/2020/handin2.pdf - 2025-05-04
Deep Learning for Nature Language Processing Deep Learning for Nature Language Processing Lianhao Yin Lund University lianhao.yin@energy.lth.se November 29, 2016 Lianhao Yin (LTH) DL for NLP November 29, 2016 1 / 21 Overview 1 Introduction 2 Word2vec 3 Recurrent Neural Networks 4 Dynamic Memory Networks Lianhao Yin (LTH) DL for NLP November 29, 2016 2 / 21 Natural Language Processing Natural langu
https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/DeepLearning/2016/NLPLianhao.pdf - 2025-05-04
Deep-Learning Study Circle: Reinforcement Learning Deep-Learning Study Circle: Reinforcement Learning Gabriel Ingesson 0/46 Reinforcement Learning The problem where an agent has to learn a policy (behavior) by taking actions in an environment, with the goal that the policy should maximize a cumulative reward. Different from supervised and unsupervised learning: No labeled training data. Reward sig
https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/DeepLearning/2016/RL.pdf - 2025-05-04
Autoencoders Autoencoders Fredrik Bagge Carlson Fredrik Bagge Carlson, Lund University: Autoencoders Introduction General idea Auto: Greek auto- "self, one’s own" Encode: from en- "make, put in" + code: a system of words, letters, figures, or symbols used to represent others Find a useful encoding, h = f(x), of data x in an unsupervised manner. Trained using an encoder h = f(x) and a decoder x̂ =
https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/DeepLearning/2016/autoencoders.pdf - 2025-05-04
Practical overview of optimization of Deep Networks Practical overview of optimization of Deep Networks Carl Åkerlindh December 15, 2016 Carl Åkerlindh | DL Training 2 / 19 Gradient descent optimization Backpropagation Batch gradient descent Online gradient descent Mini-batch gradient descent Challenges Gradient descent additions Momentum Nestrov accelerated gradient Adagrad Other SGD variants Add
https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/DeepLearning/2016/dl_optimization.pdf - 2025-05-04
