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L3: Density functions and sum-of-squares methods ○ Lyapunov Stabilization Computationally Untractable ○ Density Functions ○ “Almost” Stabilization Computationally Convex ○ Duality Between Cost and Flow ○ Sum-of-squares Optimization ○ Examples Literature. Density functions: Rantzer, Systems & Control Letters, 42:3 (2001) Synthesis: Prajna/Parrilo/Rantzer, TAC 49:2 (2004) SOSTOOLS and its Control Ap

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/NonlinearControl/2017/fu_lec03_2017eight.pdf - 2025-12-12

RLbob_4slides L5: Relaxed dynamic programming and Q-learning • Relaxed Dynamic Programming ○ Application to switching systems ○ Application to Model Predictive Control Literature: [Lincoln and Rantzer, Relaxing Dynamic Programming, TAC 51:8, 2006] [Rantzer, Relaxing Dynamic Programming in Switching Systems, IEE Proceeding on Control Theory and Applications, 153:5, 2006] Who decides the price of a

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/NonlinearControl/2017/fu_lec05_2017all.pdf - 2025-12-12

lie2017 Lecture 6 – Nonlinear controllability Nonlinear Controllability Material Lecture slides Handout from Nonlinear Control Theory, Torkel Glad (Linköping) Handout about Inverse function theorem by Hörmander Nonlinear System ẋ = f(x, u) y = h(x, u) Important special affine case: ẋ = f(x) + g(x)u y = h(x) f : drift term g : input term(s) What you will learn today (spoiler alert) New mathemat

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/NonlinearControl/2017/fu_lec06_2017nine.pdf - 2025-12-12

Synthesis, Nonlinear design ◮ Introduction ◮ Relative degree & zero-dynamics (rev.) ◮ Exact Linearization (intro) ◮ Control Lyapunov functions ◮ Lyapunov redesign ◮ Nonlinear damping ◮ Backstepping ◮ Control Lyapunov functions (CLFs) ◮ passivity ◮ robust/adaptive Ch 13.1-13.2, 14.1-14.3 Nonlinear Systems, Khalil The Joy of Feedback, P V Kokotovic Why nonlinear design methods? ◮ Linear design degra

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/NonlinearControl/2017/funonlin_lec07_synthesis_2017_eight.pdf - 2025-12-12

Session 1 Reading assignment Liberzon chapters 1 – 2.4. Exercises 1.1. = Liberzon Exercise 1.1 1.2. = Liberzon Exercise 1.5 1.3. = Liberzon Exercise 2.2 1.4. = Liberzon Exercise 2.3 1.5. Read Liberzon Chap.2.3.3 and explain how we can avoid assuming y ∈ C2. Prove Lemma 2.2 (Liberzon Exercise 2.4). 1.6. = Liberzon Exercise 2.5 (State the brachistochrone problem first.) 1.7. = Liberzon Exercise 2.6

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2018/exercise1.pdf - 2025-12-12

Session 3 Reading assignment Liberzon chapters 4.1, 4.3 – 4.5. Exercises 3.1. = Liberzon Exercise 4.1. (Deriving the Euler-Lagrange equation for brachistochrone is enough. No need to derive that its solutions are cycloids.) 3.2. = Liberzon Exercise 4.8 3.3. = Liberzon Exercise 4.10 3.4. = Liberzon Exercise 4.11 3.5. = Liberzon Exercise 4.12 3.6. = Liberzon Exercise 4.15 3.7. = Liberzon Exercise 4.

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2018/exercise3.pdf - 2025-12-12

Session 5 Reading assignment Liberzon chapter 5. Exercises 5.1. = Liberzon Exercise 5.2 5.2. = Liberzon Exercise 5.3 5.3. = Liberzon Exercise 5.4 5.4. = Liberzon Exercise 5.5 5.5. = Liberzon Exercise 5.6 5.6. = Linerzon Exercise 5.7 1

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2018/exercise5.pdf - 2025-12-12

Optimal Control 2018 Kaoru Yamamoto Optimal Control 2018 L1: Functional minimization, Calculus of variations (CV) problem L2: Constrained CV problems, From CV to optimal control L3: Maximum principle, Existence of optimal control L4: Maximum principle (proof) L5: Dynamic programming, Hamilton-Jacobi-Bellman equation L6: Linear quadratic regulator L7: Numerical methods for optimal control problems

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2018/lecture2eight.pdf - 2025-12-12

Exercise 1 1. Use MATLAB to find the generalized plant of the following diagram: Plant: G = [ 1 s , 1 s ] ∈ C1×2, controller: K = [ −1 −1 −1 1 ] , uncertainty: ∆ = [ ∆1 ∆2 ] ∈ C4×4 with ∆1,∆2 ∈ C2×2 Controller input: y = [ y1 y2 ] ∈ R2, controller ouput u ∈ R2. Controlled signal: z = [ z1 z2 ] ∈ R2 Exogenous signal: w = dr n  ∈ R3 The weighting functions Wr, Wrf , Wz1 , Wd, Wn are tunable real

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/RobustControl2025/ExercisePDF/Exercise1.pdf - 2025-12-12

Exercise 3 1. a) Let M = [ 0 A B 0 ] where A, B ∈ Cn×n are two constant square matrices. Compute the structured singular value µS(M) for S = {∆ : δI2n, δ ∈ C} S = { ∆ : ∆ ∈ C2n×2n } S = { ∆ = [ ∆1 0 0 ∆2 ] : ∆1,∆2 ∈ Cn×n } Hint : Use Schur’s formula for determinant. The results can be expressed in terms of ρ(AB), σ̄(A) and σ̄(B). b) Given a structured uncertainty S. Show that µS(M) = 1 min{σ̄(∆) |

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/RobustControl2025/ExercisePDF/Exercise3.pdf - 2025-12-12

I propose that for the exercise session the students solve the following from 'essentials of robust control' •Problem 16.5 •Problem 16.11 And also try and solve the loopshaping task in exercise 3 of the attached (the last page) using the H-infinity loopshaping method from the lecture. When solving the task rather than trying to find the w_c that minimises ||S||_\infty they should instead investiga

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/RobustControl2025/ExercisePDF/Exercise5.pdf - 2025-12-12

Robust Control Lecture 1 Dongjun Wu Course Information ~ 7 Lectures (Monday 13h15), ~ 7 exercises (Thursday 13h15) Textbooks: Essentials of Robust Control etc. Schedule and material: see course page Examination: Handins + Exam Exercises: fill in the google sheets before solving Handins are due before the exercise session, email to: dongjun.wu@control.lth.se with subject Robust control handin X Syl

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/RobustControl2025/SlidesPDF/Lec1.pdf - 2025-12-12

Robust Control Lecture 3 Dongjun Wu Uncertainty modeling Nyquist-Bode plots. e.g. for P̃ = (1+w∆)P , ∥∆∥∞ ≤ 1, choose w such that ∣∣∣∣ P̃ −P P ( jω) ∣∣∣∣≤ |w( jω)|, ∀ω. Parametric uncertainties. e.g. for ẍ + c m ẋ + k m x = F m nominal value error actual mass normalized error m m̄ ±10% m̄(1+0.1δm) |δm | ≤ 1 c c̄ ±20% c̄(1+0.2δc ) |δc | ≤ 1 k k̄ ±30% k̄(1+0.3δk ) |δk | ≤ 1 All can be put as wz ηv

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/RobustControl2025/SlidesPDF/Lec3.pdf - 2025-12-12

Generate C code for aircraft MPC problem QPgen Home Examples Installation Documentation Licence Authors Citing Generate C code for aircraft MPC problem % specify dynamics (sampled with 50 ms sampling time) MPC.Adyn = [0.9993 -3.0083 -0.1131 -1.6081; -0.0000 0.9862 0.0478 0.0000; 0.0000 2.0833 1.0089 -0.0000; 0.0000 0.0526 0.0498 1.0000]; MPC.Bdyn = [-0.0804 -0.6347;-0.0291 -0.0143; -0.8679 -0.0917

https://www.control.lth.se/fileadmin/control/Research/Tools/qpgen/gen_aircraft_mpc.html - 2025-12-12

Help file for run_code_gen QPgen Home Examples Installation Documentation Licence Authors Citing Help file for run_code_gen % ------------------------------------------------------- % [QP_reform,alg_data] = run_code_gen(QP,opts) % ------------------------------------------------------- % Generate C code to solve parametric QP:s of the form % % minimize 1/2 x'Hx + g'x + h(Cx) % subject to Ax = b %

https://www.control.lth.se/fileadmin/control/Research/Tools/qpgen/help_run_code_gen.html - 2025-12-12

Releases QPgen Home Examples Installation Documentation Licence Authors Citing Releases QPgen v0.0.2 [2014-12-19] Added features Bug fixes Efficiency improvements QPgen v0.0.1 [2014-12-11] Page generated 2015-12-09 10:07:11 CET, by jemdoc+MathJax .

https://www.control.lth.se/fileadmin/control/Research/Tools/qpgen/releases.html - 2025-12-12

Solve sequence of lasso problems QPgen Home Examples Installation Documentation Licence Authors Citing Solve sequence of lasso problems % store problem dimension n = 10000; m = 2000; % store nbr of iterations and execution times nbr_iters = []; exec_times = []; iter = 0; % run 50 different lasso problems for jj = 1:50 % aquire new data (returns random vector) g = acquire_data(m); % solve problem t

https://www.control.lth.se/fileadmin/control/Research/Tools/qpgen/solve_lasso.html - 2025-12-12

PIC-LU | Department of Automatic Control Faculty of Engineering, LTH Search Department of Automatic Control LTH, Faculty of Engineering Education Research External Engagement Personnel Publications About, Contact Home  >  Charlotta Johnsson  >  Research projects  >  Research Projects - Passed  >  PIC-LU Denna sida på svenska This page in English PIC-LU Page Manager: Mika Nishimura 2018-12-05 Sidöv

https://www.control.lth.se/personnel-old/charlotta-johnsson/research-projects/research-projects-passed/pic-lu/ - 2025-12-12

PIC-opic | Department of Automatic Control Faculty of Engineering, LTH Search Department of Automatic Control LTH, Faculty of Engineering Education Research External Engagement Personnel Publications About, Contact Home  >  Charlotta Johnsson  >  Research projects  >  Research Projects - Passed  >  PIC-opic Denna sida på svenska This page in English PIC-opic PIC-opic: Process Industry Centre - Opt

https://www.control.lth.se/personnel-old/charlotta-johnsson/research-projects/research-projects-passed/pic-opic/ - 2025-12-12

Publications 1970 – 1979 | Department of Automatic Control Faculty of Engineering, LTH Search Department of Automatic Control LTH, Faculty of Engineering Education Research External Engagement Personnel Publications About, Contact Home  >  Karl Johan Åström  >  Publications 1970 – 1979 Denna sida på svenska This page in English Publications 1970 – 1979 Books Karl Johan Åström: Introduction to stoc

https://www.control.lth.se/personnel-old/karl-johan-aastroem/publications-1970-1979/ - 2025-12-12