LG Problem Statement

This folder consists of various methods to estimate discrete-time, linear, time-varying equations. This sort of problem is called a Linear Gaussian (LG) model.

They can be characterized as the following:

$$\text{model:}{\mathbf{x}}_k={\mathbf{A}}_{k-1}{\mathbf{x}}_{k-1}+{\mathbf{v}}_k+{\mathbf{w}}_k$$ $$\text{observation model:}{\mathbf{y}}_k={\mathbf{C}}_k{\mathbf{x}}_k+{\mathbf{n}}_k$$

Where $k$ is a index in discrete time.

• ${\mathbf{x}}_k\in {\mathbb{R}}^N$ is the state of the system

• ${\mathbf{x}}_0\in {\mathbb{R}}^N\sim \mathcal{N}({\hat{\mathbf{x}}}_0,{\hat{\mathbf{P}}}_0)$ is the initial state of the system

• ${\mathbf{v}}_k\in {\mathbb{R}}^N$ input to the system. might have a mapping to ${\mathbf{v}}_k=\mathbf{B}{\mathbf{u}}_k$

• ${\mathbf{w}}_k\in {\mathbb{R}}^N\sim \mathcal{N}({\hat{\mathbf{x}}}_0,{\mathbf{Q}}_k)$ process noise

• ${\mathbf{y}}_k\in {\mathbb{R}}^N$ measurement

• ${\mathbf{n}}_k\in {\mathbb{R}}^N\sim \mathcal{N}({\hat{\mathbf{x}}}_0,{\mathbf{R}}_k)$ measurement noise

• ${\mathbf{A}}_k$ is the state transition matrix

• ${\mathbf{C}}_k$ is the observation matirx which maps our state to our measurement

The problem for state estimation is as follows: The problem of state estimation is to come up with an estimate $\widehat{{\mathbf{x}}_k}$ of the true state of a system, at one or more timesteps, $k$, given knowledge of the initial state, ${\mathbf{x}}_0$, a sequence of measurements $y_{0:K}$ a sequence of inputs ${\mathbf{v}}_{1:K}$ as well as knowledge of the system’s motion and observation models

There are roughly two paradigms to solving this:

• Bayesian Inference prior, posterior paradigm where we are updating a prior density (based on our initial state, inputs, and motion model) with our measurements to produce a posterior estimate
  • Generalized Gaussian Filter Bayes Filter Kalman Filter
• Maximum A Posteriori here we are using optimization to find the most likely posterior state given the information we have.

This problem can be solved with many approaches, one of them being the Kalman Filter

linearGaussianEstimation