Control Theory Lachlan Blackhall and Tyler Summers

Control Theory Advanced control methods are model based Use a mathematical model of the system to design controllers Controller System Dynamics uy

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

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

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

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

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

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

LQR Example Vectored thrust aircraft

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

LQR Example Equilibrium point: State space model

LQR Example Linear model

LQR Example Simulation

Automotive Many subsystems in modern cars use control principles. – hJG8A - Volvo Collision Avoidancehttp:// hJG8A – pFw&feature=related - Lexus auto parkhttp:// pFw&feature=related

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. – Google Self Driving Car –

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. – ygUhttp:// ygU

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.

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

Aeronautical (cont.) Other aeronautical systems where control is used include – cg37I – nano hummingbirdhttp:// cg37I –UAV collision avoidance –Space launch vehicles –Satellites