() Control System Design - A Perspective Bo Bernhardsson Karl Johan Åström Department of Automatic Control LTH, Lund University Bo Bernhardsson Karl Johan Åström Control System Design - A Perspective Control System Design - A Perspective 1 Introduction 2 Integrated System and Control Design 3 Control Principles 4 Requirements and Methods 5 Summary Theme: Many things to think about. Bo Bernhardsson

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() Requirements Bo Bernhardsson and Karl Johan Åström Department of Automatic Control LTH, Lund University Bo Bernhardsson and Karl Johan Åström Requirements Requirements and Limitations 1 Introduction 2 The basic feedback system 3 A broad view of control system design 4 Command signal following - System inversion 5 Disturbances 6 Process uncertainty 7 Robustness 8 Summary Theme: Requirements

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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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() 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

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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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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

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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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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

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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

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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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