Observer Design for Continuous-Time Dynamical Systems

This note is based on Pauline Bernard’s survey paper observer Design for continuous-Time dynamical systems. I highly recommend everyone who is working in the field of observer design, state estimation to read this extremely beautiful paper. She is one of the leading figures in the field of observer design and my favorite researcher in observer design.

Overview

There’s no systematic way of designing observers for nonlinear systems. But some form of nonlinear systems can be transform to a known normal form, in which an observer exists and the injectivity of the transform guarantee that we could transform it back to the original coordinate. This paper outline several normal forms of nonlinear systems based on the fundamental concepts of observability/detectability.

Problem Statement

There are three ways to define a real-time state estimation problem, given a dynamical system:

$$\begin{array}{r}\dot{x}=f(x),y=h(x)\end{array}$$

• Open loop estimation based on simulation In a probabilistic manner, ${\text{argmax}}_{x_0}P(x_0|y_{0:t},u_{0:t})$, estimate the initial state or current state via simulation assuming the model is known. In a stochastic sense it’s doing inference, while in a deterministic manner, it’s like interval estimation.
• Open loop estimation based on optimization, moving horizon optimization $\hat{x}={\text{argmin}}_{{\hat{x}}_f}{\int }_{t-\bar{t}}^t|Y({\hat{x}}_f,t;\tau )-y(\tau ){|}^{2}d\tau$, $\bar{t}\in [0,t]$ is a window parameter.
• Closed loop estimation based on dynamical systems a dynamical system $\dot{\hat{z}}=\mathcal{F}(\hat{z},y,t),\hat{x}=\mathcal{T}(\hat{z},y,t)$, which verifies ${\lim }_{t\to \infty }|\hat{x}(t)-x(t)|=0$. Here $\hat{z}=T(\hat{x})$ and $\mathcal{T}$ is the left-inverse of the transformation $T$

Remark

• The first and second approaches are possibly equivalent in the lens of inference as optimization, note that any inference problem can be rewrite as an optimization problem.
• The third approach has some connection to the optimization perspective, because the Lyapunov function can be cast as a cost function. But the mechanism under the hood is very different because an observer include the transformation from one coordinate to another and transform it back.

Detectability, Observability and Normal Forms

Distinguishability, WIP

Reference

• observer Design for continuous-Time dynamical systems