Control Theory
Lachlan Blackhall and Tyler Summers
Control Theory
• Advanced control methods are model
based
Controller
u
System
Dynamics
y
• Use a mathematical model of the
system to design controllers
State Space Models
• Inputs, outputs describe external
behavior of system
• State variables describe internal
behavior of system
• Mathematical model:
dx
x  X  Rn
 f (x,u)
dt
y  h(x,u)
u U  R m
y Y  R p
Optimal Control
• Fundamental engineering problem:
design the “best” controller given some
constraints
• Choose a function K :Y U to solve

minimize
 l(x(t),u(t))dt
0
xÝ f (x,u)
subject to

y  h(x,u)
x  X  Rn
u U  R m
y Y  R p
Optimal Control
• Optimal control problems are hard
– Infinite-dimensional, non-convex in general
• Linear quadratic problems are solvable

minimize
 (x
T
(t)Qx x(t)  u (t)Qu u(t))dt
T
0
subject to

xÝ Ax  Bu
y  Cx
x  Rn
u  Rm
y  Rp


Linear Quadratic Regulator
• Assume y = x (i.e. C = I)
• LQR = linear quadratic regulator
• Solution: optimal cost given by
1
A P  PA Qx  PBQu B P  0
T

 (x
T
T
(t)Qx x(t)  u (t)Qu u(t))dt  x (0)Px(0)
T
T
0
• Optimal controller linear in state
1
u(t)  Qu BT Px(t)
The Kalman Filter
• Often not possible to measure x directly
y  Cx  v
where v is measurement noise
• Estimate x from measurements y
(choose
 a function F :Y X )
• Solution similar to LQR
– State estimate xˆ is linear in
measurements
 y
Linear Quadratic Gaussian
• LQG = LQR + Kalman Filter

minimize
T
T
(x
(t)Q
x(t)

u
(t)Qu u(t))dt

x
0
subject to

xÝ Ax  Bu  w
y  Cx  v
x  Rn
u  Rm
y  Rp
• Optimal solution: u(t)  Qu1BT Pxˆ (t)
ˆ
from Kalman filter

x

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

Ý)  0
( , xÝ, yÝ,
u1  F1
u2  F2  m g
LQR Example
• Linear model
AT P  PA Qx  PBQu1BT P  0
u(t)  Qu1BT Px(t)
LQR Example
• Simulation
Automotive
• Many subsystems in modern cars use
control principles.
– http://www.youtube.com/watch?v=MfOgwr
hJG8A - Volvo Collision Avoidance
– http://www.youtube.com/watch?v=16Izr52l
pFw&feature=related - Lexus auto park
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
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=jtBYlwp6
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
– http://www.youtube.com/watch?v=96WePg
cg37I – nano hummingbird
– UAV collision avoidance
– Space launch vehicles
– Satellites