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Performance Analysis of Time-Delay Systems under External Perturbations Using Output-to-Output Gain
Ruslan Seifullaev, André M. H. Teixeira
TL;DR
Communication delays can affect closed-loop performance, while delayed systems require specialized analysis because they are inherently infinite-dimensional. This paper extends dissipativity-based OOG analysis using Lyapunov–Krasovskii functionals and derives delay-dependent LMI bounds, including finite-dimensional treatments for two constant-delay classes.
Problem
Communication delays may significantly affect closed-loop performance, and time-delay systems require specialized tools because their evolution depends on the entire state history over the delay interval.
Method
The paper extends the dissipativity-based OOG framework to delayed observer-based systems using Lyapunov–Krasovskii functionals, descriptor techniques, reciprocally convex inequalities, and delay-dependent LMIs.
Results
The derived conditions provide explicit upper bounds on the OOG for systems with independent measurement and actuation delays, while Padé-approximated and finite-dimension reducible constant-delay systems admit finite-dimensional OOG analysis under suitable assumptions.
Takeaways & Limitations
The framework supports performance analysis of delayed observer-based networked control systems and provides finite-dimensional treatments for two special constant-delay classes.
Takeaways & Limitations
Padé-based models are approximate, and their accuracy may deteriorate for large delays, low approximation orders, or fast transient dynamics.
Abstract
from arXiv · showhide
Communication delays are inherent in networked control systems and may significantly affect closed-loop performance. This paper investigates the impact of external perturbations in observer-based linear systems with delayed measurements and control signals using the output-to-output gain (OOG) framework. By employing dissipativity theory and Lyapunov--Krasovskii functionals, delay-dependent linear matrix inequality conditions are derived that provide explicit upper bounds on the OOG for systems with independent delays in measurement and actuation channels. In addition, two special classes of constant-delay systems are considered: Padé-approximated systems and finite-dimension reducible systems. Numerical examples illustrate the applicability of the proposed approaches
1. INTRODUCTION
Communication delays and external perturbations can degrade networked-control performance while remaining weakly visible to residual-based monitoring. The paper extends output-to-output gain analysis to delayed observer-based systems and derives tractable conditions for bounding this worst-case impact.
- Communication delays arise throughout networked control systems and can significantly affect closed-loop stability and performance.
- External perturbations may affect sensor measurements, actuator signals, or the physical process, with some remaining weakly visible to residual-based detectors.
- Existing performance and robustness metrics generally evaluate disturbance impact and detectability separately.
- Output-to-output gain quantifies the worst-case performance degradation achievable while residual output energy remains bounded.
- Prior OOG results focused primarily on finite-dimensional delay-free systems, leaving corresponding time-delay results unestablished.
- The paper derives delay-dependent LMI conditions using Lyapunov–Krasovskii functionals and studies Padé-approximated and finite-dimension reducible constant-delay systems.
2. PROBLEM FORMULATION
The paper models an observer-based networked control system with independent time-varying measurement and actuation delays, additive communication perturbations, and physical disturbances. Its two outputs characterize performance and residual behavior, enabling OOG analysis of perturbations that may keep residual energy small while degrading performance.
- The plant has state, measurement, and performance outputs, with controllability and observability assumed for the plant realization.
- Additive perturbations enter sensor and actuator channels, while a physical fault signal acts directly on the plant dynamics.
- Measurement and control signals experience bounded, slowly varying, and independently specified communication delays.
- The received measurement and actuator signals combine delayed nominal signals with additive perturbation inputs.
- The observer-based controller produces a residual output and a performance output within a delayed closed-loop model driven by external input.
- The analysis targets perturbations that keep residual energy small while significantly degrading the performance output.
- Perturbation-signal design is outside scope, so received perturbation signals are modeled directly and their exact propagation delays are treated as immaterial.
3. MAIN RESULT: PERFORMANCE ANALYSIS WITH THE OUTPUT-TO-OUTPUT GAIN
The paper formulates OOG analysis for the infinite-dimensional dynamics induced by delayed states using dissipativity and Lyapunov–Krasovskii functionals. Feasible delay-dependent LMIs then provide explicit OOG upper bounds under zero initial conditions.
- The delayed closed-loop state is represented by its entire trajectory over the maximal admissible delay interval, making the system inherently infinite-dimensional.
- OOG measures worst-case performance degradation when residual output energy remains bounded.
- The method extends output strict dissipativity from finite-dimensional storage functions to Lyapunov–Krasovskii functionals for time-delay systems.
- Under zero initial conditions, the dissipativity argument yields an explicit upper bound on OOG from the residual-energy constraint.
- The descriptor approach treats the state derivative as an additional free variable, providing extra degrees of freedom for less conservative LMI conditions.
- Reciprocally convex inequalities are used instead of Jensen’s inequality to obtain less conservative bounds for delay-dependent integral terms.
- If the derived matrix inequality is feasible, the closed-loop system satisfies the dissipativity inequality and is finite-gain L2 output-to-output stable.
4. SPECIAL CASES
For constant delays, the paper contrasts Padé-based approximate finite-dimensional models with exact finite-dimension reducible representations. Padé models support delay-free OOG analysis and perturbation construction, whereas FD-reducibility yields exact delay-free analysis under restrictive structural conditions.
- Padé approximation: Padé expansions replace constant-delay operators with rational transfer functions, producing approximate finite-dimensional delay-free models whose dimension increases with approximation order.Higher-order approximations can improve accuracy but increase system dimension.
- Padé approximation: Padé-based models enable standard delay-free OOG tools, including tractable LMI estimates and suboptimal perturbation signals for assessing practical tightness.The generalized eigenvalue formulation constructs perturbations that can be applied to the original delayed system.
- General delay-dependent analysis: The original Lyapunov–Krasovskii framework handles distinct, time-varying measurement and actuation delays without finite-dimensional reduction errors.This is why the main analysis is derived directly for the original delay system rather than relying only on Padé approximations.
- Finite-dimension reducible systems: FD-reducible systems admit an exact finite-dimensional delay-free representation after an initial delay-length transient, unlike Padé models, which remain approximate.The representation and equivalence conditions are tied to constant delays and additional structural assumptions.
- Finite-dimension reducible systems: For zero initial conditions in FD-reducible systems, both input–output maps and the OOG are independent of the delay value.The original OOG can therefore be computed directly from the finite-dimensional representation using delay-free analysis.
- Finite-dimension reducible systems: FD-reducibility is a restrictive special structural case that provides exact analysis and indicates which communication channels should be protected.Observer-based FD-reducible systems are described as naturally protected rather than vulnerable, although vulnerable configurations can arise under specific algebraic conditions.
5. NUMERICAL EXAMPLE
The numerical examples evaluate OOG bounds under independent and slowly varying communication delays, construct suboptimal perturbations, and compare delayed-system estimates with numerical gains. They also illustrate a finite-dimension reducible configuration in which admissible perturbations remain detectable without degrading performance.
- Delay-dependent OOG bounds: The delay-free system has OOG=2.007, while the delay-dependent bound increases with both measurement and actuation delays.At zero delays, the proposed framework matches the delay-free result exactly.
- Delay-dependent OOG bounds: For slowly varying equal delays, the OOG bound is governed mainly by the nominal delay, while moderate delay-rate increases have a relatively small effect.The constant-delay case therefore provides a good indication of overall behavior for slowly varying delays.
- Suboptimal perturbation signals: A generalized eigenvalue approach applied to a sampled-data approximation constructs a suboptimal perturbation signal for the delay-free system.For delayed systems, Padé realizations produce an augmented delay-free model from which a delay-adapted perturbation signal can be obtained.
- Suboptimal perturbation signals: For ℎu = ℎy = 0.1, the numerical OOG is approximately 3.08 versus an LMI-based estimate of 6.1.The comparison uses accumulated performance and residual output energies from the original delayed system under the Padé-derived perturbation.
- Suboptimal perturbation signals: For ℎu = ℎy = 0.05, the numerical OOG decreases to approximately 2.4, while the corresponding estimate is 2.95.The estimates remain informative and capture the increase in OOG associated with communication delays.
- Finite-dimension reducible system: Under finite-dimension reducibility, a measurement-channel perturbation does not affect performance output but remains visible in the residual output.This example is classified as a protected configuration because the admissible perturbation channel does not degrade performance while appearing in the residual signal.
6. CONCLUSIONS
The paper extends output-to-output gain analysis to observer-based networked control systems with communication delays and external perturbations. It derives delay-dependent LMI upper bounds and examines Padé-approximated and finite-dimension reducible constant-delay systems.
- The paper derives delay-dependent OOG upper bounds using Lyapunov–Krasovskii functionals, descriptor techniques, and reciprocally convex inequalities.
- The analysis covers independent measurement and actuation delays and two finite-dimensional constant-delay representations under suitable assumptions.