Interval Observer-Based Estimation

• Interval observer-based estimation is a framework that constructs interval-valued state estimates which guarantee the true state is contained despite uncertainties.
• It employs techniques like Metzler transformations, mixed-monotone embedding, and optimization via LP/SDP to ensure robust performance and convergence.
• Applications span from epidemiological models and privacy-preserving systems to distributed multi-agent networks, addressing uncertainties with set-membership guarantees.

Interval observer-based estimation is a framework for state estimation in dynamical systems where the goal is to construct interval-valued state estimates that are guaranteed to contain the true system state for all admissible uncertainties—including parametric uncertainties, bounded disturbances, measurement noise, and in some settings, unknown inputs or malicious attacks. Interval observers are fundamentally distinct from stochastic observers: they provide set-membership guarantees and robust bounds rather than probabilistic confidence intervals. Development of interval observers spans continuous- and discrete-time settings, linear and nonlinear dynamics, distributed/multi-agent architectures, sampled-data/impulsive systems, and hybrid, uncertain, or switched models.

1. Mathematical Principles of Interval Observers

The central goal of interval observer design is: given a dynamical system

$\dot x = f(x, u, w), \quad y = h(x, u, v)$

with $x \in \mathbb{R}^n$ (state), $u$ (known input), $w$ (unknown but bounded disturbance), $v$ (bounded measurement noise), $y$ (measured output), compute at each $t$ a pair of functions $x^-(t), x^+(t)$ such that

$$x^-(t) \le x(t) \le x^+(t) \quad \forall t,\; \forall w, v \text{ with } w \in [w^-, w^+],\; v \in [v^-, v^+]$$

with all bounds understood component-wise. These interval estimates must contract or remain bounded as $t \to \infty$, and their widths are used as metrics of observer performance.

For nonlinear systems with bounded Jacobians, mixed-monotone embedding and Jacobian sign-stable decompositions are used to construct interval observer dynamics that are “correct by construction” and ensure that the state trajectory always remains within the evolving computed intervals (Khajenejad et al., 2024, Khajenejad et al., 2022).

For linear time-varying or time-invariant systems, interval observers are constructed via coordinate transformations that convert the dynamics to positive (Metzler or nonnegative) forms (Dinh et al., 2024), enabling propagation of interval bounds under matrix splitting.

Key technical tools include:

• Positive (Metzler/nonnegative) representations to ensure comparison properties and contractivity of interval error dynamics.
• Mixed-monotone decomposition for nonlinear systems, yielding embedding systems whose interval projections contain the reachable set (Khajenejad et al., 2024, Khajenejad et al., 2022, Khajenejad et al., 14 Apr 2025).
• Affine abstractions and outer-approximation techniques for uncertain nonlinearities and partially known dynamics (Khajenejad et al., 2020, Khajenejad et al., 14 Apr 2025).
• Use of Lyapunov functions (linear or quadratic) and corresponding linear matrix inequalities (LMIs) or semidefinite programs (SDPs) for guaranteeing stability and optimizing worst-case error bounds (Aronna et al., 2017, Khajenejad et al., 2023, Khajenejad et al., 2024, Xu et al., 19 Sep 2025).

2. Interval Observer Design Methodologies

Design procedures differ according to system type, measurement model, and presence of uncertainties:

a) LTI/LTV Systems

• LTI Systems:
  • Transformation into Metzler coordinates via a Sylvester equation yields a system whose interval bounds can be propagated using positive-system splitting; when desired, a Luenberger-type structure with constant gain $L$ is used, constrained to ensure the closed-loop remains Metzler and Hurwitz (Dinh et al., 2024).
• LTV Systems:
  • Embedding into higher-dimensional target systems via invertible time-varying coordinate maps enables interval observer construction by ensuring left-invertibility and UCO conditions are met (Dinh et al., 2024).

b) Nonlinear Systems

• Mixed-monotone decomposition enables constructing embedding systems whose interval-valued states always contain the true trajectory. Tight decomposition functions, derived from sign-stable Jacobian bounds, provide minimal bounding (Khajenejad et al., 2024, Khajenejad et al., 2022, Khajenejad et al., 14 Apr 2025).
• Affine abstraction and data-driven over-approximation handle partially unknown or learned dynamics, using local and global LPs to bound nonlinearities or unknown models (Khajenejad et al., 2020).

c) Observer Gain Synthesis

• LMIs/Semidefinite Programming: LMIs encode contractivity, positivity (Metzler/nonnegative structure), and optimality conditions (e.g., $\mathcal{H}_\infty$ or $\ell_1$ gain minimization) in both linear (Khajenejad et al., 2023) and nonlinear (Khajenejad et al., 2024, Khajenejad et al., 2022, Khajenejad et al., 14 Apr 2025) cases.
• Linear Programming: For simpler cases or endpoint/face contraction in reachability-based interval observers, LPs deliver feasible observer gains (Harwood et al., 2021, Dinh et al., 2024).

d) Extensions: Switched, Impulsive, Sampled-Data, and Distributed Systems

• Impulsive/Switched/Hybrid: Interval observers for impulsive or switched systems employ infinite-dimensional LPs (relaxed via sum-of-squares techniques) to design dwell-time-constrained observer gains (Briat et al., 2017).
• Distributed and Multi-Agent: Distributed interval observer design is achieved via decentralized interval propagation and intersection, with neighbor communication and verification of collective positive detectability. Both global MILPs and local LP-based approaches are presented to guarantee network-wide stability and tightness (Khajenejad et al., 2022, Khajenejad et al., 2024).

3. Analysis of Error Dynamics, Correctness, and Performance Guarantees

For any observer, correctness is established by proving the “framing” property (the true state always remains in the computed interval):

• Positive-system comparison: The difference (interval width) system is shown to be positive, ensuring that if initialized with $x^-(0) \le x(0) \le x^+(0)$, the property is preserved for all $t$ (Aronna et al., 2017, Khajenejad et al., 2024).
• Contractivity and ISS: Lyapunov-based or comparison-based analysis yields explicit convergence/contractivity properties, rendering the observers input-to-state stable (ISS) or $\mathcal{H}_\infty$ / $L_1$ optimal with respect to disturbance widths (Khajenejad et al., 2024, Khajenejad et al., 2023).
• Ultimate boundedness: Observers operating in the presence of persistent disturbances with known bounds yield explicit formulas for the ultimate interval width, proportional to noise and modeling uncertainty (Ito, 2019, Xu et al., 19 Sep 2025).

In nonlinear settings, embedding systems and mixed-monotone decompositions ensure the interval enclosure contracts under bounded Jacobians, and tightness is achieved using optimal gain selection via MILP or MISDP formulations (Khajenejad et al., 2024).

4. Specialized Interval Observer Architectures and Applications

a) Epidemiological Models

In the time-varying SIR–SI setting, estimate-dependent interval observers with dynamically-tuned injection gains are synthesized to provide provable error bounds, despite parametric uncertainty in transmission rates. The error system is positive and monotone, with convergence established by a common linear Lyapunov function; the decay rate is epidemic-state dependent (Aronna et al., 2017).

b) Impulsive, Sampled-Data, and Switched Systems

For linear impulsive systems, range/minimum dwell-time interval observers are constructed, with gains synthesized by infinite-dimensional LPs (realized via sum-of-squares programming). These methods directly carry over to aperiodic sampled-data and continuous-time switched systems (Briat et al., 2017).

c) Privacy-Preserving and Security Applications

Interval-observer-based privacy mechanisms can deliver deterministic privacy guarantees on output signals, optimizing $\mathcal{H}_\infty$ width subject to bounded error and adversarial perturbations (Khajenejad et al., 2023). In secure CACC systems under FDI attacks, combination of model-based interval observers and real-time neural attack estimators yields certified interval state and attack estimates for resilience (Bonab et al., 18 Jan 2026).

d) Distributed Nonlinear Multi-Agent Systems

Distributed interval observer synthesis for nonlinear, bounded-error, multi-agent systems employs local and global gain design, networked interval intersection, and decomposition of bounded-error equations for both states and inputs. The theory guarantees input-to-state stability and correct-by-construction containment of adversarial or unknown inputs, even in high-dimensional networks (Khajenejad et al., 2024).

5. Extensions: Model Learning, Unknown Inputs, and Enhanced Set-Membership

Modern interval observers incorporate:

• Partial Model Learning: Data-driven outer-approximations of unknown or partially known model dynamics, with recursive refinement and affine outer-bounding, ensure the observer adapts to model uncertainty and delivers correct interval enclosures as more data are collected (Khajenejad et al., 2020, Khajenejad et al., 14 Apr 2025).
• Unknown/Adversarial Inputs and Attacks: Input/state interval observers employ mixed-monotone decompositions, auxiliary-state cancellation, and structural design to recover interval estimates for both states and unknown or strategic inputs—including, e.g., FDI attacks or unmodeled policies—under minimal prior knowledge (Khajenejad et al., 2023, Khajenejad et al., 2021).
• Measurement-Constraint Tightening: Integration of reachability-theoretic propagation with measurement-induced polyhedral constraints produces state intervals as tight as possible given all available data (Harwood et al., 2021).

6. Computational and Practical Aspects

Computational tractability is addressed via:

• Efficient per-step ODE or difference equation integration, using interval arithmetic and decomposition rules that scale quadratically (rather than exponentially) with state dimension by exploiting sign-stable decompositions (Khajenejad et al., 14 Apr 2025).
• Convex or mixed-integer optimization for gain selection, leveraging LP, SDP, MILP, or SOS relaxations (for nonlinear and impulsive systems), enabling both centralized and decentralized implementation (Briat et al., 2017, Khajenejad et al., 2024, Khajenejad et al., 14 Apr 2025).
• Statistical and set-based output metrics, including explicit formulas for ultimate interval width, convergence rates, and trade-offs between accuracy and guaranteed properties such as privacy or resilience (Khajenejad et al., 2023, Ito, 2019).

A summary comparison of key interval observer methodologies is given below:

Setting	Main Design Principle	Performance/Certainty Guarantee
LTI / LTV	Metzler/nonnegative embedding; observer gain LP or LMI	Guarantees via positivity; ultimate bounds, contractivity (Dinh et al., 2024)
Nonlinear, bounded Jacobian	Mixed-monotone embedding; multiple-gain architecture; SDP/MILP	Correct by construction; ISS/$\mathcal{H}_\infty$-optimality (Khajenejad et al., 2024, Khajenejad et al., 2022)
Partially known/learned model	Data-driven over-approximation; tight interval propagation	Simultaneous state and model entrainment; ultimate boundedness (Khajenejad et al., 2020, Khajenejad et al., 14 Apr 2025)
Distributed (LTI/nonlinear)	Local interval propagation + networked intersection; LP/MILP for gains	Network-wide tightness and ISS; scalable to large systems (Khajenejad et al., 2022, Khajenejad et al., 2024)
Privacy/security/attack resilience	Interval extension of privacy metrics; observer-based attack estimation	Certified privacy or attack detection; explicit accuracy/privacy trade-off (Khajenejad et al., 2023, Bonab et al., 18 Jan 2026, Khajenejad et al., 2021)

7. Limitations and Open Problems

While interval observer-based estimation provides strong set-membership guarantees, several limitations persist:

• Dependence on model structure: Tightness and practical feasibility depend on the ability to find Metzler/nonnegative representations (in linear systems) or tight mixed-monotone decompositions (in nonlinear settings). For some highly non-monotone or poorly observable systems, interval widths may remain large.
• Conservative bounding: For systems with large parametric uncertainty, high disturbance amplitude, or ambiguous measurement structure (e.g., unobservable periods (Aronna et al., 2017)), ultimate intervals may be conservative.
• Scalability with dimension: While local and distributed methods exist, centralized MILP/MISDP formulations can be computationally intensive in high dimensions, driving the success of distributed/local and data-driven approaches.
• Measurement tightness: Measurement constraints and noise bounds critically impact interval contraction; exactness requires known, tight bounds or robust over-approximation (Harwood et al., 2021).

Despite these issues, interval observer frameworks provide a rigorous, generalizable, and computationally efficient backbone for robust state estimation across a diverse array of uncertain, nonlinear, networked, and adversarial systems.