Your favourite restaurant in the University Centre has decided to build a new pizza-turning robot.
This new version will represent its belief that the heat lamp is
over a particular slice by a continous variable. This variable is
an angle, θ. Angle θ will be tracked by a
Kalman filter. As the robot has no sensors installed yet, the
Kalman filter will apply only the prediction step. The state
vector xt is defined as [θ, dθ/dt]T. At every timestep the control input ut
gives a constant acceleration which is added to dθ/dt.
Meanwhile, the speed of rotation decays due to friction by a
constant factor of 0.5 at every time step. The motion of the
rotating pizza is defined by the following equations:
θ = θ + Δt dθ/dt
dθ/dt = 0.5 dθ/dt + Δt ut
(Note that these equations are not particularly realistic. See
section 16.3 of "Engineering Mechanics: Statics and Dynamics" by R.C.
Hibbeler, 1992, for a more realistic model.)
Give the matrices At and Bt for the system update equation: xt = At xt-1 + Bt ut + εt.