Nonlinear Functional Estimation: Functional Detectability and Full Information Estimation

TL;DR

This work addresses estimating a nonlinear function $z_t=\phi(x_t)$ of a nonlinear time-varying system from noisy outputs when the full state is not detectable. It introduces $\delta$-IOOS as a functional detectability notion and proves it is both necessary and sufficient for the existence of a stable $\delta$-IOS functional estimator, via a Lyapunov characterization $W_{\delta}$. A full information estimation (FIE) framework is developed that, under $\delta$-IOOS, yields a $\delta$-IOS estimator; a quadratic-objective specialization provides a practical exponential ($\delta$-IOOS) design with tunable weights. The approach is demonstrated on a nonlinear power-system model where only the total load is functionally detectable, yielding accurate $z_t$ estimates while full-state estimation fails, and illustrating practical applicability and computational feasibility.

Abstract

We consider the design of functional estimators, i.e., approaches to compute an estimate of a nonlinear function of the state of a general nonlinear dynamical system subject to process noise based on noisy output measurements. To this end, we introduce a novel functional detectability notion in the form of incremental input/output-to-output stability ($δ$-IOOS). We show that $δ$-IOOS is a necessary condition for the existence of a functional estimator satisfying an input-to-output type stability property. Additionally, we prove that a system is functional detectable if and only if it admits a corresponding $δ$-IOOS Lyapunov function. Furthermore, $δ$-IOOS is shown to be a sufficient condition for the design of a stable functional estimator by introducing the design of a full information estimation (FIE) approach for functional estimation. Together, we present a unified framework to study functional estimation with a detectability condition, which is necessary and sufficient for the existence of a stable functional estimator, and a corresponding functional estimator design. The practical need for and applicability of the proposed functional estimator design is illustrated with a numerical example of a power system.

Figures (3)

• Figure 1: Ground truth states at each bus and branch flow across each line (solid lines) with corresponding measurements (dots) and FIE estimates (dashed lines).
• Figure 2: Functional value (solid green) and corresponding functional estimate resulting from our FIE approach (dashed blue) and from the simple linear estimator \ref{['eq:linear_functional_estimator']} (dashed red).
• Figure 3: Median computational time required to solve the FIE problem depending on the time step for $10$ different runs. The shaded area shows the corresponding minimal and maximal computational time.