Model Definitions #

This library provides methods for control and estimation of linear dynamical systems (LDS) of the following form: \[ \mathbf{x}_{t+1} = f\left( \mathbf{x}_{t}, \mathbf{v}_{t} \right) = \mathbf{A} \mathbf{x}_{t} + \mathbf{B} \mathbf{v}_{t} + \mathbf{m}_{t} + \mathbf{w}_{t} \]

\[ \mathbf{y}_{t} = h\left( \mathbf{x}_{t} \right) \]

t           : time index
x           : system state
v = g%u     : input (e.g., in physical units used for model fit)
u           : control signal sent to actuator (e.g., in Volts)
y           : system output
m           : process disturbance
w ~ N(0, Q) : process noise/disturbance

A           : state matrix
B           : input coupling matrix
g           : input gain (e.g., for converting to control signal actuator voltage)
              n.b., assumes this conversion is linear
Q           : process noise covariance

%           : element-wise multiplication

LDS with Gaussian Observations #

For linear dynamical systems whose outputs are assumed to be corrupted by additive Gaussian noise before measurement (Gaussian LDS models), the output function takes the following form.

\[ \mathbf{y}_{t} = \mathbf{C} \mathbf{x}_{t} + \mathbf{d} \] \[ \mathbf{z}_{t} \sim \mathcal{N}\left(\mathbf{y}_{t} , \mathbf{R} \right) \]

z           : measurement

C           : output matrix
d           : output bias
R           : measurement noise covariance

LDS with Poisson Observations #

For linear dynamical systems whose outputs are assumed to be rates underlying measured count data derived from a Poisson distribution (Poisson LDS models), the output function takes the following form. Note an element-wise exponentiation is used to rectify the linear dynamics for the rate of the Poisson process.

\[ y_{t}^{i} = \exp \left(\mathbf{c}^i \mathbf{x}_{t} + d^i\right) \] \[ z_{t}^i \sim \rm{Poisson} \left(y_{t}^i \right) \]

i           : output index

z           : measurement (count data)

c           : i^th row of output matrix (C)
d           : output bias