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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 +

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/ControlSystemsSynthesis/2016/handin1.pdf - 2025-11-11

() 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

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/ControlSystemsSynthesis/2016/handin3.pdf - 2025-11-11

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

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/ControlSystemsSynthesis/2016/handin5.pdf - 2025-11-11

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

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/ControlSystemsSynthesis/2016/lqg.html - 2025-11-11

Vertical Aerospace Dynamics, Maciejowski example 5.8 References Maciejowski pp. 244-258, Plots on LQGLTR designs Maciejowski pp. Appendix pp 405-406, Describes the model. Hung and MacFarlane, Multivariable Feedback: A quasi-classical Approach , Lecture Notes in Control and Information Sciences, vol 40, Springer-Verlag The problem is to control the vertical-plane dynamics of an aircraft. There are

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/ControlSystemsSynthesis/2016/mac58.html - 2025-11-11

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

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/ControlSystemsSynthesis/2016/mill.html - 2025-11-11

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

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/ControlSystemsSynthesis/2016/tgen.html - 2025-11-11

Session 2 Dissipativity and Integral Quadratic Constraints Reading assignment You don’t need to read everything from these papers, but check the main results and some examples. Jan C. Willems was the leading figure of systems and control in the Netherlands for several decades. The other two papers are from our department. • Jan C. Willems, Arch. Rational Mech. and Analysis, 45:5 (1972). • A. Megre

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/NonlinearControl/2017/2017_E2.pdf - 2025-11-11

Session 4 Hybrid systems Reading assignment Check the main results and examples of these papers. • Johansson/Rantzer, IEEE TAC, 43:4 (1998). • Chizeck/Willsky/Castanon, Int. J. on Control, 43:1 (1986) Exercise 4.1Consider two pendula[ ẋ1 ẋ2 ] = [ x2 1− x1 ] [ ẋ1 ẋ2 ] = [ x2 −1− x1 ] which are swinging around x1 = 1 and x1 = −1 respectively. a. Find a control law that brings the state to (0, 0)

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/NonlinearControl/2017/2017_E4.pdf - 2025-11-11

Session 6 Nonlinear Controllability Reading assignment • Glad, Nonlinear Control Theory, Ch. 8 + Hörmander handout Exercises marked with a “*” are more difficult Exercise 6.1 Consider a car with N trailers. The front-wheels of the car can be controlled, and the car can drive forwards and backwards. Describe a manifold that can be used as state-space. Show that its dimension is N + 4. Exercise 6.2

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/NonlinearControl/2017/2017_E6.pdf - 2025-11-11

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, backstepping, forwarding Exercise sessions: Solve 50% of prob

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/NonlinearControl/2017/fu_lec02_2017eight.pdf - 2025-11-11

L4: Hybrid systems and dynamic programming • Hybrid Systems ○ Piecewise Linear Systems ○ Piecewise Quadratic Lyapunov Functions ○ Value Iteration ○ Policy Iteration ○ Jump Linear Systems Literature: Piecewise Quadratic: Johansson/Rantzer, IEEE TAC, 43:4 (1998) Networked Control Example: Nilsson/B/W, Automatica 34:1 (1998) Value and policy iteration: web.mit.edu/dimitrib/www/Det_Opt_Control_Lewis_V

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/NonlinearControl/2017/fu_lec04_2017eight.pdf - 2025-11-11

Session 2 Reading assignment Liberzon chapters 2.5 – 3. Exercises 2.1. = Liberzon Exercise 2.9. Reading the discussion in Chapter 1.2.2 will be helpful. 2.2. = Liberzon Exercise 2.10 2.3. = Liberzon Exercise 2.14 2.4. = Liberzon Exercise 3.2 2.5. = Liberzon Exercise 3.5 2.6. = Liberzon Exercise 3.6 for k = 1 and n = 2. 2.7. = Liberzon Exercise 3.8 1

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2018/exercise2.pdf - 2025-11-11

Session 4 Reading assignment Liberzon chapter 4.2. Exercises 4.1. = Liberzon Exercise 4.2 4.2. = Liberzon Exercise 4.3 4.3. = Liberzon Exercise 4.4 4.4. = Liberzon Exercise 4.5 4.5. = Liberzon Exercise 4.6 1

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2018/exercise4.pdf - 2025-11-11

Session 6 Reading assignment Liberzon chapter 6. Exercises 6.1. = Liberzon Exercise 6.1 6.2. = Liberzon Exercise 6.2 6.3. = Liberzon Exercise 6.3 6.4. = Liberzon Exercise 6.4 6.5. = Liberzon Exercise 6.5 6.6. = Linerzon Exercise 6.6 1

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2018/exercise6.pdf - 2025-11-11

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 L4: Maximum principle, Existence of optimal control L5: Dynamic programming, Hamilton-Jacobi-Bellman equation L6: Linear quadratic regulator L7: Numerical methods for optimal control problems Exercise

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2018/lecture1eight.pdf - 2025-11-11

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/lecture3eight.pdf - 2025-11-11

Optimal Control 2018 LionSealGrey Optimal Control 2018 Yury Orlov 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

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2018/lecture4.pdf - 2025-11-11

Exercise 2 1. Consider a nominal plant P (s) = 1 s+ 1 . and a set of perturbed models Π = {P̃ : P̃ = (1 + w∆)P, ∆ ∈ H∞, ∥∆∥∞ ≤ 1} in which w = 0.125s+ 0.25 (0.125/4)s+ 1 . Find the extreme parameter values in each of the plants (a) - (g) below so that each plant belongs to the set Π. All parameters are assumed to be positive. (a) Neglected delay: find the largest θ for Pa = Pe−θs; (b) Neglected la

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/RobustControl2025/ExercisePDF/Exercise2.pdf - 2025-11-11

Exercise 4 1. Consider the system e e K P W M ? 7 7 7 7 g ? 7 7 7 r uw u e − − where M = 4 s2 + 2s+ 4 , P (s) = 10(s+ 2) (s+ 1)3 , W (s) = 0.1(s+ 1) s+ 10 . a) Find the H2-optimal controller from r to (e, uw). Repeat with W = ϵ for ϵ = 0.01 and 0.0001. Study the behavior of the controller when ϵ → 0. b) Repeat a) by replacing H2 by H∞. 2. Consider the system e eK P W2 ∆ W1 7 7 7 o o oow g 7 7 7 7

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/RobustControl2025/ExercisePDF/Exercise4.pdf - 2025-11-11