Proximal observers for secure state estimation

TL;DR

This work addresses online secure state estimation for discrete-time nonlinear systems subject to sparse impulsive measurement noise. It introduces proximal adaptive observers that perform prediction followed by a measurement update computed by minimizing a convex (potentially nonsmooth) loss, encapsulated as hat{x}_t = argmin_z [ 1/2 ||W_t^{-1}(z - hat{x}_{t|t-1})||^2 + g_t(z) ] with g_t(z) = ψ(V_t^{-1}(y_t − C_t z)). Proximal operators enable efficient online updates and yield nonlinear error dynamics, with stability and convergence results established in linear and nonlinear settings, including Kalman-like specialization and component-wise relaxations that furnish closed-form updates. The paper develops several robust instantiations, notably ℓ1-based and Lasso-type losses, and demonstrates through linear and nonlinear simulations that the estimators achieve resilience to arbitrarily large impulsive noise while offering tractable online computation. The theoretical framework hinges on bounded weighting matrices, observability notions, and convex analysis to guarantee convergence in the noise-free case and boundedness under impulsive disturbances, highlighting practical applicability to cyber-physical systems facing adversarial data integrity threats. Overall, the proposed proximal robust observers provide a flexible, online, optimization-driven approach to secure state estimation with practical update rules and demonstrable robustness advantages.

Abstract

This paper discusses a general framework for designing robust state estimators for a class of discrete-time nonlinear systems. We consider systems that may be impacted by impulsive (sparse but otherwise arbitrary) measurement noise sequences. We show that a family of state estimators, robust to this type of undesired signal, can be obtained by minimizing a class of nonsmooth convex functions at each time step. The resulting state observers are defined through proximal operators. We obtain a nonlinear implicit dynamical system in term of estimation error and prove, in the noise-free setting, that it vanishes asymptotically when the minimized loss function and the to-be-observed system enjoy appropriate properties. From a computational perspective, even though the proposed observers can be implemented via efficient numerical procedures, they do not admit closed-form expressions. The paper argues that by adopting appropriate relaxations, simple and fast analytic expressions can be derived.

Figures (5)

• Figure 1: Averaged norms (over $100$ realizations) of the absolute estimation errors when only the sparse measurement is active. Minimum dwell-time between consecutive occurrences of nonzero instances of the sparse noise is fixed $5$ samples (component-wise).
• Figure 2: Averaged norms, over $100$ realizations, of the (absolute) estimation errors when only the sparse measurement is active.
• Figure 3: Averaged norms of the absolute estimation errors (over $100$ realizations) when all process and measurement noises are active. A zoom on the interval $\interval{450}{500}$.
• Figure 4: Averaged norms of the (absolute) estimation errors when all process and measurement noise are active over $100$ realizations.
• Figure 5: Nonlinear example -- Averaged norms of the (absolute) estimation errors when all process and measurement noise are active over $100$ realizations.

Theorems & Definitions (1)

• Definition 1: Generalized UCO