ex01.dvi Exercise Session 1 1. To evaluate a controlled system the maximum values of the sensitivity function and the complementary sensitivity functions have been computed giving max ω |S(iω)| = 2.45, max ω |T (iω)| = 1.70 Use these numbers to estimate the largest amplification of disturbances that may occur. Also provide an estimate of the precision in the transfer function required for the clos

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ex3.dvi Exercise Session 3 1. Describe your results on Handin 2. 2. a) Show that state feedback control u = −Lx̂ + lryr, where x̂ is given by a Kalman filter, can be written as U(s) = −Cfb(s)Y (s) + Cff (s)Yr(s) with Cfb(s) = L(sI − A + BL − KC)−1K Cff (s) = (I − L(sI − A + BL − KC)−1B)lr = (I + L(sI − A + KC)−1B)−1lr b) Show that the controller above can be written as R(s)U = −S(s)Y + T (s)Yr wit

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ex5.dvi Exercise 5 BottomUp and Interaction 1. Explain why the standard Smith predictor does not work for processes with integration or unstable dynamics. 2. Smith’s controller for a process P(s) = P0(s)e −sL with time delay is given by C(s) = C0(s)Cpred(s), Cpred(s) = 1 1+ P0(s)C0(s)(1− e−sL) where C0 is the nominal controller for the process P0 without delay and L is the time delay. The transfer

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() Handin 4 A) Conditional integration are methods where windup is avoided by suspending integration under certain circumstances, for example when the error is large or when the control signal saturates. Construct a counterexample which shows that such methods may result in systems that have equilibria with nonzero error. (Thanks to F. Bagge-Carlsson for raising this question) B) Suggest a scheme

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Landaus Flexible Transmission References Hand-out, 4 pages Landau et al, The Combined Pole Placement/ Sensitivity Shaping Method , Internal Report Grenoble, 1994 The problem is to design a SISO controller for a flexible transmission. The same controller should work for three drift cases (0, 50 and 100%). There are several specifications. It is hard to meet all of them simultaneously. Matlab-code T

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() Control System Design - LQG Bo Bernhardsson, K. J. Åström Department of Automatic Control LTH, Lund University Bo Bernhardsson, K. J. Åström Control System Design - LQG Lecture - LQG Design Introduction The H2-norm Formula for the optimal LQG controller Software, Examples Properties of the LQ and LQG controller Design tricks, how to tune the knobs What do the “technical conditions” mean? How to

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() Control System Design - LQG Part 2 Bo Bernhardsson, K. J. Åström Department of Automatic Control LTH, Lund University Bo Bernhardsson, K. J. Åström Control System Design - LQG Part 2 Lecture - LQG Design What do the “technical conditions” mean? Introducing integral action, etc Loop Transfer Recovery (LTR) Examples For theory and more information, see PhD course on LQG Reading tip: Ch 5 in Macie

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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

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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

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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

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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

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Monotone Operators and Fixed-Point Iterations Monotone Operators and Fixed-Point Iterations Pontus Giselsson 1 Today’s lecture • operators and their properties • monotone operators • Lipschitz continuous operators • averaged operators • cocoercive operators • relation between properties • monotone inclusion problems • special case: composite convex optimization • resolvents and reflected resolvent

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IntroductionDeep Learning - Study Circle Deep Learning - Study Circle Bo Bernhardsson, Kalle Åström, Magnus Fontes, Fredrik Bagge Carlsson, Martin Karlsson Agenda Intro by me, Fontes FredrikB Kalle all Decide weekly meeting date Decide upon the first topics and responsible About the course Engineering perspective Hands on experience and intuition Use existing material Structure 1-2 persons respons

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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 - 2025-07-11

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)

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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)

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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

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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

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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

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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

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