Linear Systems Theory FRT001F 9p Description Mathematical theory for linear dynamical systems. Topics covered • Linear time-varying (LTV) systems. • Linearisation. • Coordinate transformation. • Linear time-periodic (LTP) systems. • Floquet-decomposition. • Controllability and observability. • Gramians. • Controllable and observable form. • Minimal realisation. • Internal and Lyapunov stability. •

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Session 0 Math background Reading Assignment Get the book. If you cannot get a hard copy, here is the link for an e-copy (Link expired by Dec. 2019): https://www.dropbox.com/s/03tsrmwmvsik5w5/Rugh_Linear%20System%20Theory. pdf?dl=0 Exercise 0.1 Compute eAt for A = [ 1 2 0 3 ] . Exercise 0.2 Given a positive semi-definite and symmetric matrix A ∈ Rn×n where only one of its eigenvalues is 0 and all

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LionSealWhite Lecture 6 Least squares problems Adjoint operators 1 / 32 LionSealWhite Review: Least Squares Solution to Linear Equations (I) Consider a system of linear equations Ax = b, A ∈ Rm×n, b ∈ Rm with m ≥ n and rank(A) = n (Tall A—more rows than columns, or more equations than unknowns). If b /∈ range(A) then the linear system is inconsistent, i.e., no solution exists. Find x that minimize

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Adaptive Control Bo Bernhardsson and K. J. Åström Department of Automatic Control LTH, Lund University Bo Bernhardsson and K. J. Åström Adaptive Control Adaptive Control 1 Introduction 2 Model Reference Adaptive Control 3 Recursive Least Squares Estimation 4 The Self-Tuning Regulator 5 Real Adaptive Controllers 6 Summary Bo Bernhardsson and K. J. Åström Adaptive Control Introduction Adapt to adjus

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IQC toolbox IQC toolbox Gustav Nilsson May 25, 2016 IQC - Integral Quadratic Constraints • A unifying framework for systems analysis • Generalizes stability theorems such as small gain theorem and passivity theorem • Generalizes many concepts from robust control analysis • (Fairly) easy to build computer tools (convex optimization) Outline • Some theory on IQC • IQCβ toolbox • Live demo ICQ - Theo

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() Pole Placement Design Bo Bernharsson and Karl Johan Åström Department of Automatic Control LTH, Lund University Bo Bernharsson and Karl Johan Åström Pole Placement Design Pole Placement Design 1 Introduction 2 Simple Examples 3 Polynomial Design 4 State Space Design 5 Robustness and Design Rules 6 Model Reduction 7 Oscillatory Systems 8 Summary Theme: Be aware where you place them! Bo Bernharss

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() Robust Control, H∞, ν and Glover-McFarlane Bo Bernharsson and Karl Johan Åström Department of Automatic Control LTH, Lund University Bo Bernharsson and Karl Johan Åström Robust Control, H∞, ν and Glover-McFarlane Robust Control 1 MIMO performance 2 Robustness and the H∞-norm 3 H∞-control 4 ν-gap metric 5 Glover-MacFarlane Theme: You get what you ask for! Bo Bernharsson and Karl Johan Åström Rob

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ACC 1990 Benchmark Example References ACC 1990, pp. 961-962, The problem ACC 1990, pp. 963-973, Several Solutions ACC 1991, pp. 1931-1932, Mats Liljas design and comments The problem is to design a controller for a scalar linear system consisting of two-mass spring system. There are 4 states.

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Aircraft dynamics with wind gust turbulence References Anderson and Moore p 222 The problem is to damp out the turbulence giving forward and aft acceleration on an aircraft. The model is of order 6. There is one input: rudder position. Matlab-code Model and code in fig822.m .

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ex4.dvi Exercise 4 Poleplacement and PID 1. Use Euclid’s algorithm to find all solutions to the equation 7x+ 5y = 6 where x and y are integers. 2. Use Euclid’s algorithm to find all solutions to the equation s2 x(s) + (0.5s+ 1)y(s) = 1 where x(s) and y(s) are polynomials. Use the results to find a solution to the equation s2 f (s) + (0.5s+ 1)(s) = (s2 + 2ζcω cs+ω2 c)(s 2 + 2ζoωos+ω2 o) such that t

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ex6.dvi Exercise 6 LQG and H∞ 1. Use the appropriate Riccati equation to prove the Kalman filter identity R2 + C2(sI − A)−1 R1(−sI − AT)−1CT 2 = [Ip + C2(sI − A)−1 L]R2[Ip + C2(−sI − AT)−1 L]T Use duality to deduce the return difference formula Q2 + BT(−sI − AT)−1Q1(sI − A)−1B = [Im + K(−sI − AT)−1B]T Q2[Im + K(sI − A)−1B] 2. Consider the Doyle-Stein LTR example from the LQG lecture G(s) = s+ 2 (s

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Extremum-seeking Control Extremum-seeking Control Tommi Nylander and Victor Millnert May 25, 2016 1 / 14 Short introduction I Non-model based real-time optimization I When limited knowledge of the system is available I E.g. a nonlinear equilibrium map with a local minimum I Popular around the middle of the 1950s I Revival with proof of stability 1 I Very attractive with the increasing complexity o

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() Gain Scheduling Bo Bernhardsson and Karl Johan Åström Department of Automatic Control LTH, Lund University Bo Bernhardsson and Karl Johan Åström Gain Scheduling Gain Scheduling What is gain scheduling ? How to find schedules ? Applications What can go wrong ? Some theoretical results LPV design via LMIs Conclusions To read: Leith & Leithead, Survey of Gain-Scheduling Analysis & Design To try ou

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() Handin 1 Bo Bernhardsson, K. J. Åström Department of Automatic Control LTH, Lund University Bo Bernhardsson, K. J. Åström Handin 1 Handin 1 - goals Get some practice using the Matlab control system toolbox (or similar) Get started with some control design Bo Bernhardsson, K. J. Åström Handin 1 Example - Double Integrator Consider the double integrator y = 1 s2 u controlled with state-feedback +

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() Handin 3 Consider the (broomstick) system p2 s2 − p2 with p = 6 rad/s ((1 feet). Hint: You might find it useful to read or watch Gunter Stein’s Bode Lecture. a) Find a stabilizing controller achieving pT(iω)p < (Ωa/ω) 2, when ω > Ωa = 10 rad/s Ms := max ω pS(iω)p < 10 b) Try to get as low Ms you can, while maintaining the requirement on T. Bonus: Try to find a theoretical lower bound on Ms (the

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handin5.dvi Handin 5 - Connected Inverted Pendulums (LQG) x1 x2 φ1 = x5 φ2 = x7 u1 u2 The process consists of two inverted pendulums mounted on movable carts. The carts are connected with a spring. The inputs are the forces on the two carts. The outputs are the cart positions and pendulum angles. The system hence have 2 inputs and 4 outputs. The system parameters correspond to 1m pendulums mounted

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LQG-examples exkj2.m Slow process pole example fig822.m Aircraft turbulence attenuation lqg1.m An example where some eigenvalues are moved by LQG but some others are fixed. lqg2.m An example technical conditions are violated lqg3.m LTR example Doyle and Stein AC 79 exreducedobserver.m Reduced order design mac58.m LTR design example from Maciejowski at the end of lecture 10

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Thickness Control of a Rolling Mill References Lars Malcom Pedersen's Lic-thesis The problem is to design a controller for the rolling mill at the "danska stalverket". There are two inputs, the signals to the hydraulic valves at the north and south side. The output is a vector describing the (predicted) thickness profile of the plate. There are 6 states. Matlab-code Description mill.ps The model m

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Turbo-generator model TGEN References Mac App. A.2. Two input two output 6 state model of a large turbo-alternator. Hung and MacFarlane (1982), "Multivariable Feedback: A Quasi-classical Approach" Lecture Notes in Control and Information Sciences, vol 40, Springer Verlag. Matlab-code The model tgen.m

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