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Fundamental Limitations in MIMO Systems Fundamental Limitations in MIMO Systems M.T Andrén J. Berner Control System Synthesis, 2016 M.T Andrén, J. Berner Fundamental Limitations in MIMO Systems Control System Synthesis, 2016 1 / 21 Outline 1 Some concepts Singular values Pole and zero directions Sensitivity functions 2 Bode’s Integral Theorem 3 RHP Poles & Zeros Interpolation Constraints Specifi
() Mini Lectures and Projects Bo Bernhardsson and Karl Johan Åström Department of Automatic Control LTH, Lund University Bo Bernhardsson and Karl Johan Åström Mini Lectures and Projects Mini Lectures Last part of the examination 1-2 persons 15 minutes presentations Decide topic before April 25 Middle of May, date on home page Bo Bernhardsson and Karl Johan Åström Mini Lectures and Projects Suggest
https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/ControlSystemsSynthesis/2016/minilectures.pdf - 2026-04-26
Mixed H/H2-synthesis and Youla-parametrization Mixed H∞/H2-synthesis and Youla-parametrization Olof Troeng 2016-05-25 Motivation (1/2) Control of electric field in accelerator cavity. Very simple process P(s) = 1 1 + sT e−sτ , Optimal controller? : P(I)(D), LQG, Smith Predictor, (MPC) Inspiration from (Garpinger 2009). Motivation (1/2) Control of electric field in accelerator cavity. Very simple p
Discrete time mixed H2 / H control Discrete time mixed H2/H∞ control Yang Xu Department of Automatic Control Lund University May 25, 2016 Introduction Continuous time mixed H2/H∞ control problem: ◮ Zhou, Kemin, et al. ”Mixed H2 and H∞ performance objectives. I. Robust performance analysis.” Automatic Control, IEEE Transactions on 39.8 (1994): 1564-1574. ◮ Doyle, John, et al. ”Mixed H2 and H∞ perfo
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 - 2026-04-26
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 - 2026-04-26
Session 3 Density functions and sum-of-squares methods Reading assignment Check the main results and examples of these papers. • Rantzer, Systems & Control Letters, 42:3 (2001). • Prajna/Parrilo/Rantzer, TAC 49:2 (2004). • SOSTOOLS and its Control Applications, Prajna/P/S/P (2005) Exercise 3.1 a. Draw a phase plot for the system[ ẋ1 ẋ2 ] = [ −εx1 + x2 1 − x2 2 −εx2 + 2x1x2 ] for ε = 1. b. Prove
https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/NonlinearControl/2017/2017_E3.pdf - 2026-04-26
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 - 2026-04-26
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 - 2026-04-26
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 - 2026-04-26
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 - 2026-04-26
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 - 2026-04-26
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 - 2026-04-26
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 - 2026-04-26
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 - 2026-04-26
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 - 2026-04-26
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 - 2026-04-26
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 - 2026-04-26
