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Rootlocus Rootlocus Gautham Department of Automatic Control 1/8 Gautham: Rootlocus Rootlocus Method (Rotortmetoden) Plotting of the root locus 2/8 Gautham: Rootlocus The Rootlocus Method(Rotortmetoden) Introduction I Graphical method of solving algebraic equations introduced by Walter R.Evans. in 1948. I Instead of solving equations for fixed values of parameters, the equation is solved for all va
Lateral Dynamics of Aeroplane References Anderson, Moore, Optimal Control, Linear quadratic methods, 2nd ed , Prentice Hall 1990, Sec 6.2 Harvey and Stein, Quadratic Weights for Regulator Properties , IEEE AC 1978, pp 378-387 Friedland, Control System Design , pp. 40-47. Nice description of Aerodynamics for control The problem is to design a state feedback controller u = -Lx. There are two input s
https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/ControlSystemsSynthesis/2016/steinflyg.html - 2026-01-17
Session 1 — Readings and exercises limit cycles, existence/uniqueness, Lyapunov, regions of attraction Reading assignment Khalil Chapter 1–3.1, (not 2.7), 4–4.6 Comments on chapter 2.6 The main topic is about existance of periodic orbits for planar systems and the most important subjects are the Poincaré-Bendixson Criterion and the Bendixson Criterion. Lemma 2.3 and Corollary 2.1 can also be used
https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/NonlinearControl/2017/2017_E1.pdf - 2026-01-17
Session 5 Relaxed dynamic programming and Q-learning Reading assignment Check the main results and examples of these papers. • Lincoln/Rantzer, TAC 51:8 (2006) • Rantzer, IEE Proc on Control Theory and Appl. 153:5 (2006) • Geramifard et.al, Found. & Trends in Machine Learn. 6:4 (2013) Exercise 5.1Consider the linear quadratic control problem Minimize ∞∑ t=0 x(t)2 + u(t)2 subject to x(t+ 1) = x(t)
https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/NonlinearControl/2017/2017_E5.pdf - 2026-01-17
Reading instructions and problem set 7 Feedback linearization, zero-dynamics, Lyapunov re-design, backtep- ping Reading assignment Khalil [3rd ed.] Ch 13. Khalil [3rd ed.] Ch.14.(1) 2-4 + "The joy of feedback" by P. Kokotović (handout) (Extra reading: • “Constructive Nonlinear Control” by R. Sepulchre et al, Springer, 1997) • “Nonlinear & Adaptive Control Design” by M. Krstić et al, Wiley, (1995)
https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/NonlinearControl/2017/2017_E7.pdf - 2026-01-17
Nonlinear Control Theory 2017 Anders Rantzer m.fl. Nonlinear Control Theory 2017 L1 Nonlinear phenomena and Lyapunov theory L2 Absolute stability theory, dissipativity and IQCs L3 Density functions and computational methods L4 Piecewise linear systems, jump linear systems L5 Relaxed dynamic programming and Q-learning L6 Controllability and Lie brackets L7 Synthesis: Exact linearization, backsteppi
https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/NonlinearControl/2017/fu_lec01_2017eight.pdf - 2026-01-17
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 - 2026-01-17
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 - 2026-01-17
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 - 2026-01-17
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
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 - 2026-01-17
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 - 2026-01-17
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 - 2026-01-17
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 - 2026-01-17
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 - 2026-01-17
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 - 2026-01-17
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 - 2026-01-17
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 - 2026-01-17
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 - 2026-01-17
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 - 2026-01-17
