Compensation of state-dependent input delay for nonlinear systems
Abstract
We introduce and solve stabilization problems for linear and nonlinear systems with state-dependent input delay. Since the state dependence of the delay makes the prediction horizon dependent on the future value of the state, which means that it is impossible to know a priori how far in the future the prediction is needed, the key design challenge is how to determine the predictor state. We resolve this challenge and establish closed-loop stability of the resulting infinite-dimensional nonlinear system for general non-negative-valued delay functions of the state. Due to an inherent limitation on the allowable delay rate in stabilization of systems with time-varying input delays, in the case of state-dependent delay, where the delay rate becomes dependent on the gradient of the delay function and on the state and control input, only regional stability results are achievable. For forward-complete systems, we establish an estimate of the region of attraction in the state space of the infinite-dimensional closed-loop nonlinear system and for linear systems we prove exponential stability. Global stability is established under a restrictive Lyapunov-like condition, which has to be a priori verified, that the delay rate be bounded by unity, irrespective of the values of the state and input. We also establish local asymptotic stability for locally stabilizable systems in the absence of the delay. Several illustrative examples are provided, including unicycle stabilization subject to input delay that grows with the distance from the reference position.
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System (1)-(3) is no longer equivalent to the classical time-delay system in [5,6,14-17].
“Equation (31) indicates that system (1)-(3) is no longer equivalent to the following classical time-delay system in [5,6,14-17]”
From Multi-layer predictor feedback design for nonlinear integro-differential equations with state-dependent input delays (2026) · cites this paper as [6]
The controller incorporates a compensator that effectively cancels the influence of inhomogeneous terms in the PDE actuator state, without imposing restrictions on the severity of the nonlinearity or the feasibility condition, while guaranteeing the latter through an assumption on the positivity of the transport speed to avoid an ill-posed problem.
“Our controller incorporates a compensator that effectively cancels the influence of inhomogeneous terms in the PDE actuator state, without imposing any restrictions on the severity of the nonlinearity or the feasibility condition [5,6], while guaranteeing the latter through an assumption on the positivity of the transport speed to avoid an illposed problem (see [14,17]).”
From Multi-layer predictor feedback design for nonlinear integro-differential equations with state-dependent input delays (2026) · cites this paper as [6]
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- Authors
- Bekiaris-Liberis, N., & Krstic, M.
- Affiliations
- University of California San Diego (US)
- Venue
- IEEE Transactions on Automatic Control
- Published
- 2013-02
- Volume
- 58
- Issue
- 2
- Pages
- 275-289
- ISSN
- 0018-9286 (ISSN-L)
- 1558-2523
- OpenAlex
- W2022076727
- Cited by
- 229
- References
- 60
- Citations by year
- 2026: 4 · 2025: 9 · 2024: 4 · 2023: 14 · 2022: 22 · 2021: 16 · 2020: 12 · 2019: 13 · 2018: 19 · 2017: 28 · 2016: 22 · 2015: 26 · 2014: 19 · 2013: 18 · 2012: 3
- FWCI
- 30.56
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- IEEE Transactions on Automatic Control (publishedVersion)
- Topics
- Physical Sciences › Engineering › Control and Systems Engineering › Stability and Control of Uncertain Systems
- Physical Sciences › Engineering › Control and Systems Engineering › Advanced Control Systems Optimization
- Physical Sciences › Computer Science › Computer Networks and Communications › Neural Networks Stability and Synchronization
- Keywords
- time-varying input delay, nonlinear systems, linear systems, closed-loop stability, regional stability, domain of attraction, exponential stability, local asymptotic stability
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