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L4: Hybrid systems and dynamic programming • Hybrid Systems ○ Piecewise Linear Systems ○ Piecewise Quadratic Lyapunov Functions ○ Value Iteration ○ Policy Iteration ○ Jump Linear Systems Literature: Piecewise Quadratic: Johansson/Rantzer, IEEE TAC, 43:4 (1998) Networked Control Example: Nilsson/B/W, Automatica 34:1 (1998) Value and policy iteration: web.mit.edu/dimitrib/www/Det_Opt_Control_Lewis_V

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/NonlinearControl/2017/fu_lec04_2017eight.pdf - 2026-01-15

Session 2 Reading assignment Liberzon chapters 2.5 – 3. Exercises 2.1. = Liberzon Exercise 2.9. Reading the discussion in Chapter 1.2.2 will be helpful. 2.2. = Liberzon Exercise 2.10 2.3. = Liberzon Exercise 2.14 2.4. = Liberzon Exercise 3.2 2.5. = Liberzon Exercise 3.5 2.6. = Liberzon Exercise 3.6 for k = 1 and n = 2. 2.7. = Liberzon Exercise 3.8 1

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2018/exercise2.pdf - 2026-01-15

Session 4 Reading assignment Liberzon chapter 4.2. Exercises 4.1. = Liberzon Exercise 4.2 4.2. = Liberzon Exercise 4.3 4.3. = Liberzon Exercise 4.4 4.4. = Liberzon Exercise 4.5 4.5. = Liberzon Exercise 4.6 1

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2018/exercise4.pdf - 2026-01-15

Session 6 Reading assignment Liberzon chapter 6. Exercises 6.1. = Liberzon Exercise 6.1 6.2. = Liberzon Exercise 6.2 6.3. = Liberzon Exercise 6.3 6.4. = Liberzon Exercise 6.4 6.5. = Liberzon Exercise 6.5 6.6. = Linerzon Exercise 6.6 1

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2018/exercise6.pdf - 2026-01-15

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 L4: Maximum principle, Existence of optimal control L5: Dynamic programming, Hamilton-Jacobi-Bellman equation L6: Linear quadratic regulator L7: Numerical methods for optimal control problems Exercise

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2018/lecture1eight.pdf - 2026-01-15

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/lecture3eight.pdf - 2026-01-15

Optimal Control 2018 LionSealGrey Optimal Control 2018 Yury Orlov 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

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2018/lecture4.pdf - 2026-01-15

Exercise 2 1. Consider a nominal plant P (s) = 1 s+ 1 . and a set of perturbed models Π = {P̃ : P̃ = (1 + w∆)P, ∆ ∈ H∞, ∥∆∥∞ ≤ 1} in which w = 0.125s+ 0.25 (0.125/4)s+ 1 . Find the extreme parameter values in each of the plants (a) - (g) below so that each plant belongs to the set Π. All parameters are assumed to be positive. (a) Neglected delay: find the largest θ for Pa = Pe−θs; (b) Neglected la

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/RobustControl2025/ExercisePDF/Exercise2.pdf - 2026-01-15

Exercise 4 1. Consider the system e e K P W M ? 7 7 7 7 g ? 7 7 7 r uw u e − − where M = 4 s2 + 2s+ 4 , P (s) = 10(s+ 2) (s+ 1)3 , W (s) = 0.1(s+ 1) s+ 10 . a) Find the H2-optimal controller from r to (e, uw). Repeat with W = ϵ for ϵ = 0.01 and 0.0001. Study the behavior of the controller when ϵ → 0. b) Repeat a) by replacing H2 by H∞. 2. Consider the system e eK P W2 ∆ W1 7 7 7 o o oow g 7 7 7 7

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/RobustControl2025/ExercisePDF/Exercise4.pdf - 2026-01-15

Robust Control Lecture 2 Dongjun Wu Uncertainties Dynamic (frequency-dependent) uncertainties. Unmodeled dynamics at high frequency (phase completely unknown at high frequencies!) Imperfect measurements ⇒ uncertain inputs. Nonlinearities. Parametric uncertainties. Inaccurate description of components. Variations of system parameters. LFT General framework: wz ηv uy 7 ? 7 ? ? ? K ∆ P ℓLFT and uLFT:

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/RobustControl2025/SlidesPDF/Lec2.pdf - 2026-01-15

Robust Control Lecture 4 Dongjun Wu Structured uncertainty Uncertainty structure S =  ∆=  δ1I . . . δs I ∆1 . . . ∆r  : δi ∈C, ∆ j ∈Cm j×n j  , Uncertainty set: ∆ ∈ S, ∥∆∥∞ ≤ 1. Robust stability structured uncertainty Theorem (RS test for structured complex uncertainty) M and ∆ are ::::: stable :::::: without :::::: hidden :::::::: un

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/RobustControl2025/SlidesPDF/Lec4.pdf - 2026-01-15

Generate code for lasso problem QPgen Home Examples Installation Documentation Licence Authors Citing Generate code for lasso problem % set problem dimensions q = 2000; n = 10000; % generate sparse data matrix F = sprandn(q,n,10/n); % regenerate until all columns are non zero while not(isempty(find(sum(F,2) == 0))) F = sprandn(q,n,10/n); end % store data in QP struct QP.H = F'*F; QP.G = F'; QP.C =

https://www.control.lth.se/fileadmin/control/Research/Tools/qpgen/gen_lasso.html - 2026-01-15

Help file for run_code_gen_MPC QPgen Home Examples Installation Documentation Licence Authors Citing Help file for run_code_gen_MPC % ------------------------------------------------------- % [QP_reform,alg_data] = run_code_gen_MPC(MPC,opts) % Actions: % - reformulates MPC problem in struct MPC to QP problem in struct QP % - runs: run_code_gen(QP,opts) % -------------------------------------------

https://www.control.lth.se/fileadmin/control/Research/Tools/qpgen/help_run_code_gen_MPC.html - 2026-01-15

Simulate aircraft MPC system QPgen Home Examples Installation Documentation Licence Authors Citing Simulate aircraft MPC system % nbr of time steps nbr_steps = 100; % create reference trajectory r_y1 = 0.25; r_y2 = 10; g1 = repmat(MPC.Q*[0;-r_y2;0;-r_y2],10,1); g1 = [g1;zeros(20,1)]; g2 = zeros(60,1); % time instance when reference is changed change_ref = 50; % initial condition X0 = zeros(4,1); %

https://www.control.lth.se/fileadmin/control/Research/Tools/qpgen/sim_aircraft_mpc.html - 2026-01-15

Pauline Kergus – Pauline Kergus Born November 25, 1992 H +33 (0)6 68 87 95 87 B pauline.kergus@control.lth.se Education 2016–2019 PhD in Automatic Control, ONERA, Toulouse, France. Supervision: Charles Poussot-Vassal and Fabrice Demourant. Title: "Data-driven model reference control in the frequency-domain: From model reference selection to controller validation." Thesis available at tel-3084374 2

https://www.control.lth.se/fileadmin/control/staff/PaulineKergus/CV_KERGUS.pdf - 2026-01-15

LISA2 | 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  >  LISA2 Denna sida på svenska This page in English LISA2 Future manufacturing system need to be more flex

https://www.control.lth.se/personnel-old/charlotta-johnsson/research-projects/research-projects-passed/lisa2/ - 2026-01-15

Curriculum Vitae | 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  >  Curriculum Vitae Denna sida på svenska This page in English Curriculum Vitae Curriculum Vitae Karl J. Åström was born in Östersund, Sweden on Augu

https://www.control.lth.se/personnel-old/karl-johan-aastroem/curriculum-vitae/ - 2026-01-15

Publications before 1970 | 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 before 1970 Denna sida på svenska This page in English Publications before 1970 Books Karl Johan Åström: Reglerteori . Almqvi

https://www.control.lth.se/personnel-old/karl-johan-aastroem/publications-before-1970/ - 2026-01-15

Misc | 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  >  Pontus Giselsson  >  Misc Denna sida på svenska This page in English Misc Software QPgen : A C code generator for quadratic optimization problems Pendulum Movies Time-optimal path

https://www.control.lth.se/personnel-old/pontus-giselsson/misc/ - 2026-01-15

Automotive Systems | 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  >  Rolf Johansson  >  Automotive Systems Denna sida på svenska This page in English Automotive Systems Research Publications of Prof. Rolf Johansson   Publications in Au

https://www.control.lth.se/personnel-old/rolf-johansson/automotive-systems/ - 2026-01-15