Computationally Efficient State and Model Estimation via Interval Observers for Partially Unknown Systems

TL;DR

The paper tackles interval observer design for partially unknown nonlinear systems with bounded disturbances, aiming to jointly estimate states and learn a model for the unknown dynamics. It combines Jacobian sign-stable ($\mathrm{JSS}$) decompositions, tight mixed-monotone decomposition, and data-driven over-approximations to construct interval framers that enclose the true augmented state $z_k=[x_k^\top d_k^\top]^\top$; correctness and tightening of bounds are proven as more data is gathered. An SDP-based framework is developed to synthesize observer gains that achieve $\mathcal{H}_{\infty}$-optimal performance and input-to-state stability (ISS) of the framer error, accounting for the unknown dynamics via data-driven abstractions with provable bounds $\underline{h}_k(z_k) \le h(z_k) \le \overline{h}_k(z_k)$. A representative example demonstrates that the proposed method achieves substantial computational efficiency compared to prior work, enabling real-time robust state and model estimation in safety-critical settings. The approach provides a principled, scalable path to robust interval estimation in partially known nonlinear systems, with potential extensions to time-varying and hybrid dynamics.

Abstract

This paper addresses the synthesis of interval observers for partially unknown nonlinear systems subject to bounded noise, aiming to simultaneously estimate system states and learn a model of the unknown dynamics. Our approach leverages Jacobian sign-stable (JSS) decompositions, tight decomposition functions for nonlinear systems, and a data-driven over-approximation framework to construct interval estimates that provably enclose the true augmented states. By recursively computing tight and tractable bounds for the unknown dynamics based on current and past interval framers, we systematically integrate these bounds into the observer design. Additionally, we formulate semi-definite programs (SDP) for observer gain synthesis, ensuring input-to-state stability and optimality of the proposed framework. Finally, simulation results demonstrate the computational efficiency of our approach compared to a method previously proposed by the authors.

Figures (3)

• Figure 1: Actual states, $x_{1,k}$, $x_{2,k}$, as well as their corresponding upper and lower framers outputted by the proposed observer \ref{['eq:observer']}, $\overline{x}_{1,k}$, $\underline{x}_{1,k}$, $\overline{x}_{2,k}$, $\underline{x}_{1,k}$ in a horizon of $K=2000$ time steps.
• Figure 2: Actual state, $x_k$, as well as its estimated framers, outputted by the proposed observer \ref{['eq:observer']}, $\overline{x}_{1,k}$, $\underline{x}_{1,k}$, $\overline{x}_{2,k}$, $\underline{x}_{2,k}$ in addition to the state framers obtained from the approach in khajenejad2021modelstate, $\overline{x}^{\text{MZ}}_{1,k}$, $\underline{x}^{\text{MZ}}_{1,k}$, $\overline{x}^{\text{MZ}}_{2,k}$, $\underline{x}^{\text{MZ}}_{2,k}$, in a horizon of $K=250$ time steps.
• Figure 3: Actual unknown dynamics function $h(\zeta)$, its upper and lower framer intervals $\overline{h}_{k},\underline{h}_{k}$ at time step $k=250$, as well as its global abstraction $A^h\zeta+\overline{e}^h,A^h\zeta+\underline{e}^h$ via khajenejad2021modelstate at the initial step.

Theorems & Definitions (15)

• Definition 1: Interval
• Proposition 1
• Definition 2: Lipschitz Continuity
• Definition 3: Jacobian Sign-Stability
• Proposition 2: JSS Decomposition
• Definition 4
• Proposition 3: Tight Mixed-Monotone Decomposition Functions
• Definition 5: Correct Interval Framers and Model Resilient $\mathcal{H}_{\infty}$-Optimal Interval Observer
• Lemma 1: Tractable & Tight Bounds for Data-Driven Abstractions
• proof