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# Control Theory

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1. Control Theory Lachlan Blackhall and Tyler Summers

2. Control Theory • Advanced control methods are model based • Use a mathematical model of the system to design controllers u System Dynamics y Controller

3. State Space Models • Inputs, outputs describe external behavior of system • State variables describe internal behavior of system • Mathematical model:

4. Optimal Control • Fundamental engineering problem: design the “best” controller given some constraints • Choose a function to solve minimize subject to

5. Optimal Control • Optimal control problems are hard • Infinite-dimensional, non-convex in general • Linear quadratic problems are solvable minimize subject to

6. Linear Quadratic Regulator • Assume y = x (i.e. C = I) • LQR = linear quadratic regulator • Solution: optimal cost given by • Optimal controller linear in state

7. The Kalman Filter • Often not possible to measure x directly where v is measurement noise • Estimate x from measurements y (choose a function ) • Solution similar to LQR • State estimate is linear in measurements y

8. Linear Quadratic Gaussian • LQG = LQR + Kalman Filter minimize subject to • Optimal solution: from Kalman filter

9. LQR Example • Vectored thrust aircraft

10. LQR Example • Equations of motion (Newton’s Laws) • Nonlinear! • We can linearize any nonlinear system about an equilibrium point

11. LQR Example • Equilibrium point: • State space model

12. LQR Example • Linear model

13. LQR Example • Simulation

14. Automotive • Many subsystems in modern cars use control principles. • http://www.youtube.com/watch?v=MfOgwrhJG8A - Volvo Collision Avoidance • http://www.youtube.com/watch?v=16Izr52lpFw&feature=related - Lexus auto park

15. Automotive (cont.) • DARPA Challenge • Two challenges. • The first to drive unaided across the desert. • The second to drive unaided around a city while performing a number of common tasks like parking. • http://www.youtube.com/watch?v=BSS0MZvoltw • Google Self Driving Car • http://www.youtube.com/watch?v=64w-v-RJpk8

16. Aeronautical • Aircraft have been an obvious candidate for control systems given the complexity of these systems. • Autopilots are a obvious example. • Preventing the Dutch roll mode when landing was solved using a control system called a yaw damper. • http://www.youtube.com/watch?v=jtBYlwp6ygU

17. Aeronautical (cont.) • Traditionally, the performance (manoeuvrability, etc…) and handling of an aircraft were limited by the stability properties of an aircraft. • Modern control systems have solved this fundamental problem ensuring stability but allowing high performance. • Modern fighter aircraft are actually unstable. A human pilot can no longer control the plane but a control system can make the system stable and high performance.

18. Aeronautical (cont.) • Modern aircraft now have fly-by-wire control systems that include: • Autopilot • Yaw dampers • Vibration damping • Auto-landing • Flutter prevention

19. Aeronautical (cont.) • Other aeronautical systems where control is used include • http://www.youtube.com/watch?v=96WePgcg37I – nano hummingbird • UAV collision avoidance • Space launch vehicles • Satellites