Prescribed-Time Observer

• Prescribed-time observers are estimation techniques that drive error to zero exactly at a preset time, regardless of initial conditions.
• They employ time-varying gains—such as the scaling function μ(t, T)—to compress error trajectories and robustly reject both matched and unmatched disturbances.
• Validated by rigorous Lyapunov analysis and simulations, these observers are crucial for applications in robotics, aerospace, and distributed networked control.

A prescribed-time observer is a dynamic estimation mechanism that drives the estimation error to zero within a designer-specified, finite, and non-asymptotic time window, regardless of the initial conditions. In contrast to classical finite-time or asymptotic observers, prescribed-time observers guarantee convergence at an exact, user-selected terminal time. This property is crucial for safety-critical and time-sensitive applications in advanced control, cooperative robotic systems, and networked cyber-physical systems, where strict timing and uniform performance guarantees are required.

1. Foundations and Principle of Prescribed-Time Observation

The core methodology underlying prescribed-time observers is the introduction of time-varying gains whose singularities are strategically “placed” at the prescribed convergence instant $T$. Typically, a scaling function $\mu(t, T)$ is defined as

$\mu(t, T) = \frac{T}{T-t},$

which increases monotonically and diverges as $t \to T^-$. The observer dynamics incorporate these scaling factors to accelerate the error decay, effectively “compressing” the trajectory of an otherwise infinite-time stable observer into $[0, T)$ (Abdelhadi et al., 20 Sep 2025, Shakouri et al., 2021, Ye et al., 2022).

For an $n$th order system in triangular or canonical form, the general observer design is: $$\begin{aligned} & \dot{\hat{x}}_i = \hat{x}_{i+1} + \hat{f}_i + L_i \mu^{i(1+m)}(t, T) \big(y - \hat{x}_1\big),\quad 1 \leq i < n \ & \dot{\hat{x}}_n = \hat{f}_0 + L_n \mu^{n(1+m)}(t, T) \big(y - \hat{x}_1\big) \end{aligned}$$ with $L_i$ selected so that $A - LC$ is Hurwitz and $m > 0$ (Abdelhadi et al., 20 Sep 2025, Heij et al., 4 Feb 2025, Ye et al., 2022). The observer prompts all error components to converge to zero at $t = T$ for arbitrary initial uncertainty.

2. Robustness Against Disturbances and Uncertainties

Prescribed-time observer designs are structurally robust with respect to bounded (potentially large) disturbances and unmodeled system dynamics. The Lyapunov-based analysis shows that, under the scaling induced by $\mu(t, T)$, both matched and certain classes of unmatched disturbances are completely rejected as the observer gain increases (Abdelhadi et al., 20 Sep 2025).

Specifically, for a nonlinear system under an additive disturbance $d(t)$ and possibly unmodeled perturbations, the transformed error dynamics via

$$z(t) = \Gamma e(t), \qquad \Gamma = \mathrm{diag}\left\{\frac{1}{\mu^{1+m}}, \ldots, \frac{1}{\mu^{n(1+m)}}\right\}$$

ensure that for appropriately chosen $L_i$ and $m$, the error trajectory diminishes to zero at precisely $t = T$, irrespective of the disturbance magnitude, provided it is bounded. The analysis distinguishes between contraction regions and transient regimes, confirming that the prescribed-time observer enacts a non-conservative and globally robust error rejection mechanism.

3. Comparison with High-Gain Observers and Transient Peaking

Traditional high-gain observers (HGOs) use a static large gain ($\sim\alpha_i/\epsilon$), accelerating convergence but risking pronounced transient “peaking” and noise amplification. The prescribed-time formulation, through its strictly time-varying and state-scaled gain, eliminates persistent peaking. Simulation comparisons demonstrate that for the same initial estimation error:

• Standard HGOs produce large overshoots in the observer state trajectory.
• Prescribed-time observers feature a controlled, monotonic error decay profile, with lower error magnitudes throughout transient and rapid settling at the deadline (Abdelhadi et al., 20 Sep 2025).

Simulation figures substantiate the marked reduction in transient peaking and improved accuracy for prescribed-time observers, including improved steady-state estimation and simultaneous, accurate state and disturbance estimation.

4. Mathematical Design and Lyapunov Analysis

The convergence of the prescribed-time observer is established via a time-dependent Lyapunov function,

$V(z(t)) = z(t)^T P z(t),$

where $P > 0$, and the derivative along the observer error dynamics satisfies

$$\dot{V} \leq -\lambda_1 \mu^{1+m}(t, T)\|z\|^2 + a \mu^{1+m}(t, T)\|z\|^3 + b\|z\|.$$

By stratifying the analysis into zones inside and outside contraction balls and leveraging the scaling property of $\mu(t,T)$, global uniform prescribed-time error convergence is obtained. Gains are selected so the negative term dominates any positive contributions from Lipschitz nonlinearities and bounded disturbances.

The design principle extends naturally across various nonlinear system configurations, including block-triangular and observable canonical forms, and is equally applicable to distributed observer networks (Heij et al., 4 Feb 2025).

5. Distributed and Networked Prescribed-Time Observation

In distributed and multi-agent contexts, prescribed-time observers are constructed through time-varying local output injection gains and consensus protocols. Each agent uses only its local measurements and information from neighbors (via Laplacian-weighted state differences) to estimate the global state (Heij et al., 4 Feb 2025, Ye et al., 25 Apr 2024, Zuo et al., 16 Jul 2024).

The observer dynamics in distributed form: $$\dot{z}^i = A z^i + B \varphi(z^i) + \Gamma(t) L^i \big(y^i - H^i z^i\big) - \mu^{1+m}(t) r_i \sum_j a_{ij} [z^i - z^j],$$ guarantee that state estimation errors converge to zero at the prescribed time, across all agents, even when each agent's local observability is limited and only local measurements are available.

These approaches are validated by rigorous Lyapunov arguments, linear matrix inequalities (LMIs) for gain selection, and simulations demonstrating error convergence to zero at the specified deadline. The architecture is essential for synchronous estimation in formation control, cooperative tracking, and networked control systems requiring global observability from partial, distributed sensing.

6. Practical Implications, Applications, and Implementation

Prescribed-time observers are particularly critical in applications where estimation deadlines must be met with strict guarantees:

• Robotics and safety-critical automation, where sensing delays can compromise safety or task success.
• Aerospace and automotive systems under heavy disturbances.
• Distributed sensor networks with limited bandwidth, where initialization and recalibration can be tightly scheduled.

The main limitations are the requirement for time-varying gains to diverge at $t \to T$, which may pose hardware or numerical challenges, particularly in the final approach to $T$. Practical implementations may employ gain saturation, switching strategies (transitioning to frozen-time constant gains before $T$, as in the frozen-time eigenvalue approach (Shakouri, 2021)), or PPTA functions (saturated, $C^1$-continuous prescribed-time adjustment functions) to circumvent numerical singularities without sacrificing convergence guarantees (Shao et al., 6 Apr 2025).

7. Demonstrated Performance and Summary Figures

Simulation results across varied platforms—single or multi-agent nonlinear systems, systems with disturbance and measurement uncertainties, and under severe unmodeled dynamics—demonstrate:

• Exact or near-exact convergence of the state and disturbance estimate at time $T$.
• Strong suppression of transient peaking compared to high-gain designs.
• Maintenance of small estimation errors even when exposed to large, rapidly changing perturbations.

In one key demonstration, for a nonlinear system with disturbance $d(t) = 5\sin(2t)$, both the state and the disturbance are estimated simultaneously with high fidelity and prescribed-time exactness. The performance persists across distributed observer networks with strongly connected communication topologies.

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Prescribed-time observers represent a class of estimation schemes that enforce strict temporal convergence by internalizing singular time-varying gains, providing natural robustness properties, compatibility with nonlinear and networked architectures, and marked improvements over high-gain observers in terms of transient behavior and disturbance attenuation. Their rigorous mathematical construction and strong application-driven performance have established them as a practical and theoretically well-founded tool in time-critical state and disturbance estimation (Abdelhadi et al., 20 Sep 2025, Heij et al., 4 Feb 2025, Shakouri, 2021, Shao et al., 6 Apr 2025).