Sampled-data control under time-varying delays: a robust approach for high-renewable smart grids

Abstract

The increasing reliance on inverter-based renewable energy sources in smart grids makes closed-loop stability highly sensitive to communication-induced uncertainties, including time-varying delays, sampling jitter, and packet loss. Conventional sampled-data and delay-dependent controllers typically address these impairments in isolation or rely on conservative worst-case designs, limiting their effectiveness under dynamically changing communication quality. This paper proposes a robust adaptive sampled-data control framework that explicitly links communication degradation to control-layer adaptation while preserving tractable stability guarantees. A bounded delay–jitter intensity index, , is introduced as an online-measurable proxy for communication quality and is used to schedule the feedback gain in real time. Stability is rigorously certified using a delay-weighted Lyapunov–Krasovskii functional and affine Linear Matrix Inequality (LMI) conditions verified at the admissible uncertainty endpoints, ensuring exponential stability under the combined effects of delay, jitter, and packet loss. The proposed approach is validated on a hybrid renewable microgrid with inverter-based distributed energy resources under stochastic communication impairments. Across multiple scenarios—including bounded delay, high jitter, and 10% packet loss–the adaptive controller reduces settling time by up to 33%, overshoot by 52%, and control-related energy cost by 40% compared to fixed-gain and worst-case robust baselines. In addition, cyber-aware operational reliability metrics confirm consistent preservation of admissible operating margins under degraded communication conditions. These results position the proposed method as a stability-certified control-layer complement to cyber-resilient and data-driven smart grid architectures, enabling reliable operation of high-renewable grids under realistic communication constraints.

Introduction

The integration of renewable energy sources (RES), such as solar and wind, into modern electrical grids has significantly transformed conventional power systems into highly complex smart grid infrastructures. These smart grids promise enhanced sustainability, improved reliability, and greater efficiency in power management. However, the intermittent and unpredictable nature of renewable generation presents critical operational challenges. In particular, smart grids that heavily rely on inverter-based renewable resources require advanced control methodologies capable of accommodating rapid fluctuations in supply and demand while ensuring stable and efficient grid operation.

Beyond physical variability, modern smart grids are increasingly characterized as cyber–physical systems, where sensing, communication, and control layers are tightly coupled with power system dynamics. In such settings, the reliability and resilience of grid operation are strongly influenced by the quality and availability of communication networks. Communication-induced uncertainties–such as time-varying delays, sampling jitter, and data packet losses–can act as cyber–physical stressors that degrade control effectiveness, reduce stability margins, and compromise reliable operation under high renewable penetration.These communication-induced effects are also widely recognized in synchrophasor-based WAMS/WAPAC deployments, where latency and data availability constraints can limit the reliability of wide-area monitoring and control; in this work, analogous phenomena are addressed at the control layer for inverter-based microgrids. In practice, smart-grid control loops are exposed not only to benign network variability, but also to cybersecurity and ICT integration challenges that affect availability and timeliness of measurements and commands. Congestion, misconfiguration, routing instabilities, and cyber events such as denial-of-service, jamming, or compromised network elements can manifest at the control interface as increased latency, sampling irregularity, and intermittent data unavailability. Such cyber imperfections reduce effective stability margins and can propagate from the communication layer to the control layer and ultimately to operational reliability, motivating ICT–power system co-design approaches in which communication quality and control robustness are jointly addressed. In this work, we do not claim attack detection; rather, we model these cyber-relevant manifestations through time-varying delay, jitter, and packet loss and design a stability-certified adaptive sampled-data controller that preserves safe operating margins under degraded ICT conditions, consistent with smart-grid cybersecurity guidance and reliability-oriented ICT requirements^1–3.

A key challenge in this context arises from the discrete nature of sampling in digital control systems, commonly referred to as sampled-data control. This approach relies on discrete-time sensor measurements and actuator commands transmitted over communication networks, inherently exposing the control loop to time-varying delays, irregular sampling intervals, and packet losses. These network-induced imperfections can severely degrade system performance and stability, rendering traditional continuous-time or fixed-delay control methods inadequate for communication-constrained smart grid environments.

Consequently, there is an urgent need for robust sampled-data control strategies that explicitly account for cyber–physical communication uncertainties and their impact on grid stability and reliability. The key distinction of the present work is that communication degradation is not treated as a fixed worst-case bound or as an auxiliary uncertainty handled implicitly in the analysis. Instead, delay and sampling jitter are explicitly quantified online through a bounded scalar index , which directly schedules the feedback gain while preserving a tractable Lyapunov–Krasovskii stability certificate. This enables a communication-quality–driven adaptation mechanism that bridges timing information and control action, rather than relying on plant-dependent gain scheduling or conservative fixed-gain designs. Addressing this challenge requires control frameworks that not only guarantee stability under ideal conditions, but also adapt dynamically to degraded communication quality encountered in practical smart grid operation. This paper proposes a novel robust adaptive sampled-data control framework explicitly tailored for high-renewable smart grid applications operating under realistic communication constraints. The primary objective is to maintain stability and acceptable performance in the presence of time-varying delays, sampling jitter, and packet losses, thereby supporting reliable cyber–physical grid operation.

This study is intentionally positioned as a control-layer complement to (i) data-driven and learning-based smart-grid methods focused on prediction, estimation, or policy learning, (ii) cyber-security studies emphasizing detection, identification, or mitigation of attacks, and (iii) reliability frameworks that quantify risk and service continuity under ICT contingencies. In contrast, the novelty here is a stability-certified sampled-data controller that explicitly quantifies communication degradation via the delay intensity index and uses it to adapt the feedback gain in real time. Accordingly, this work does not propose intrusion detection, attack classification, or cyber-defense mechanisms; rather, it provides a guaranteed-stability control mechanism that preserves bounded behavior and operational margins (e.g., SOP/EIR) when ICT conditions degrade.This study does not introduce new Lyapunov–Krasovskii theory, new LMI convexification methods, or a fundamentally new sampled-data control paradigm. Instead, the contribution is an application-driven and communication-aware integration of established delay-dependent tools into a stability-certified gain-scheduled sampled-data controller for inverter-dominated microgrids. In particular, the scalar index is used as a bounded, online-measurable proxy of communication degradation (delay and jitter) to schedule the gain while preserving convex LMI verification over the admissible uncertainty set. Importantly, the controller structure, stability conditions, and simulation results reported in this manuscript remain unchanged; only the contribution positioning and rationale are clarified.

Contrast with classical delay-dependent gain scheduling

Classical delay-dependent designs typically compute a single stabilizing gain for a prescribed worst-case delay bound (yielding conservative nominal performance), or they schedule gains directly with respect to the delay value (often via delay gridding, polytopic interpolation, or switching among delay regions), while assuming near-periodic sampling and treating jitter/packet loss separately. In contrast, the proposed formulation schedules the feedback gain using a bounded communication-quality proxy that aggregates delay and sampling irregularity into a single normalized measure of timing degradation. Because both the gain and the associated LMI stability conditions depend affinely on , stability over the full admissible communication envelope is certified by endpoint feasibility checks at , avoiding delay–jitter gridding and scenario-specific redesign. This yields aggressive nominal behavior when communication is healthy and automatically introduces conservatism only during delay/jitter bursts, while remaining compatible with packet-loss modeling through .

The main contributions are as follows: (i) a communication-aware sampled-data modeling framework that captures the combined effects of time-varying delay, sampling jitter, and packet loss in inverter-based microgrid control loops; (ii) a stability-certified gain-scheduled sampled-data controller in which a bounded online indicator schedules the feedback gain , and exponential (mean-square) stability is verified via endpoint LMI feasibility over the admissible uncertainty set; (iii) a cyber-aware operational reliability assessment (SOP/EIR) that maps communication degradation to control-layer performance and operational margin violations under Monte Carlo network realizations.

What is fundamentally enabled by the proposed formulation

The proposed formulation enables communication-quality–driven gain adaptation with a tractable stability guarantee, by embedding a bounded, online-measurable timing-degradation index directly into both the feedback law and the Lyapunov–Krasovskii stability conditions. The affine dependence of and the associated LMIs on allows robust stability to be certified over the entire admissible communication envelope via endpoint feasibility checks, without delay–jitter gridding or scenario-specific redesign. This enables fast nominal performance while preserving stability during transient communication degradation.

Literature review

The development of sampled-data control systems has received significant attention in renewable-dominated smart grids, particularly for addressing stability challenges introduced by digital control and communication constraints. Existing smart-grid control studies under communication constraints can be broadly grouped into several streams: (i) sampled-data and delay-dependent stabilization methods, (ii) event-triggered and communication-aware control strategies aimed at reducing network usage, (iii) robust, , and gain-scheduled designs addressing worst-case uncertainty, (iv) predictive and model-based control approaches incorporating delay bounds, and (v) cyber–physical reliability and data-driven frameworks that assess the operational impact of ICT degradation. The following review discusses these streams in turn, emphasizing their underlying assumptions and limitations with respect to joint delay, jitter, and packet-loss handling. In renewable-rich inverter-based microgrids, the unresolved challenge is not communication delay, sampling jitter, or packet loss in isolation, but their combined impact on closed-loop stability margins and operational reliability under time-varying communication quality. Accordingly, this review synthesizes these categories and critiques existing studies with respect to joint delay–jitter–loss handling and stability-certified control adaptation.

Delay-dependent and sampled-data stabilization methods

Early foundations of sampled-data control were established through delay-dependent stability analysis. Fridman⁴ proposed a refined input-delay approach that enabled rigorous Lyapunov-based stability conditions for sampled-data systems, forming a theoretical basis for subsequent developments. Extending these ideas to nonlinear and uncertain settings, Gunasekaran and Joo⁵ introduced a free-weight-matrix-based fuzzy sampled-data control framework, while Sakthivel et al.⁶ investigated robust reliable sampled-data control for stochastic systems with random delays. These works provide strong delay-dependent guarantees but typically assume bounded or fixed sampling behavior.

More recent studies incorporated stochasticity and disturbances into sampled-data formulations. Shen et al.⁷ analyzed sampled-data systems with stochastic delays and disturbances and derived LMI-based stability conditions. In power-system-oriented applications, Zhang and Hredzak⁸ proposed a distributed control framework for microgrids that accommodates aperiodic sampling and time-delayed data for batteries and renewable energy sources. Structural aspects of sampled-data control were further explored by Lu et al.⁹, who examined controllability of multi-agent systems with switching topologies under sampled-data constraints.

Robust and gain-scheduled control approaches under uncertainty

Robust control approaches have also been applied to renewable-dominated power systems. Lam et al.¹⁰ employed multivariable control for primary frequency regulation in photovoltaic-rich microgrids, demonstrating robustness against system uncertainties. In parallel, synchrophasor-based monitoring and control motivated studies on communication performance, with the U.S. Department of Energy highlighting the role of synchrophasor technologies for grid reliability¹¹. Empirical analysis of PMU reporting latency further confirmed that communication delays and missing data directly affect protection and control performance¹², which is reflected in synchrophasor data transfer standards¹³. Communication-constrained designs for wide-area protection were subsequently shown to influence achievable reliability margins¹⁴.

Power-electronic and inverter-focused sampled-data control designs

Beyond protection, sampled-data control has been applied to power electronic interfaces. Almér et al.¹⁵ investigated sampled-data model predictive control for voltage source inverters, highlighting performance trade-offs under discrete-time implementation. He and Wang¹⁶ addressed dynamic output-feedback sampled-data control under variable delays, providing stability results relevant to networked power systems. Distributed and asynchronous sampled-data strategies were further examined by Wang et al.¹⁷, who studied tracking in heterogeneous networks, and by Shanmugam et al.¹⁸, who proposed decentralized sampled-data control for interconnected power systems with stochastic disturbances.

Model-free and application-oriented designs have also been reported. Boubakir et al.¹⁹ developed a robust model-free controller for grid-connected photovoltaic systems, while Luo and Hu²⁰ analyzed stability of sampled-data load frequency control systems with multiple delays. Practical implementations under communication imperfections were further discussed by Yesudhas et al.²¹, who considered fuzzy dissipative sampled-data control for wind turbine systems with packet losses and delays. Theoretical robustness of asynchronous sampled-data networks was addressed by Xiao et al.²², and sampled-data load frequency control was experimentally explored by Guau et al.²³.

Predictive, observer-based, and data-driven control strategies

Observer-based and predictive strategies have complemented control design efforts. Zaggaf et al.²⁴ proposed sampled-data observer design for sensorless wind energy systems, while Shahzad et al.²⁵ reviewed model predictive control strategies in microgrids. Data-driven stability and stabilization of sampled-data load frequency control were investigated by Fan et al.²⁶, and adaptive sampled-data automatic generation control was studied by Ma et al.²⁸. Broader optimization perspectives on time-delay control were surveyed by Wu et al.²⁷.

Cyber–physical resilience, reliability, and ICT-aware control frameworks

From a resilience and cybersecurity standpoint, Gupta and Sharma²⁹ examined grid resilience using sampled-data-based robust controllers, while Zhang et al.³⁰ designed resilient sampled-data control under denial-of-service attacks. Fundamental sampling properties under delay were revisited by Ou et al.³¹. Delay-tolerant resilient model predictive control for microgrids under cyber–physical attacks was addressed by Liu et al.³². Emerging adaptive and learning-based approaches have also been proposed, including direct adaptive control via data-enabled policy optimization³³, distributionally robust model predictive control for smart grids³⁴, Mohan et al. proposed a distributed intrusion detection system (DIDS) for SCADA-based smart grids that employs semantic-based rules to identify cyber-attacks by correlating operational context with network events, thereby improving detection accuracy in distributed control environments³⁵ and federated learning-based policy optimization³⁶.

Fuzzy sampled-data stabilization techniques were further developed by Lee and Park³⁷. In parallel, learning-based energy management and cyber-resilience frameworks gained attention. Du and Li³⁸ applied deep reinforcement learning for multi-microgrid energy management, while Jin et al.³⁹ proposed software-defined networking architectures for cyber-resilient microgrids. Attack modeling and detection were explored by Presekal et al.⁴⁰, and reliability impacts of communication and cyber layers were analyzed by Lawal and Teh⁴¹].Composite reliability of synchrophasor-supported systems under communication availability constraints was quantified by Jimada-Ojuolape and Teh^44,45, while recent work examined transmission capacity prediction and PMU cybersecurity in data-driven and cyber–physical contexts^46,47. Recent studies have further addressed the resilience of load frequency control under cyber threats, including multi-point false data injection attacks and denial-of-service attacks, by developing detection, defense, and dissipativity-based integral sliding-mode control strategies that enhance robustness against coordinated cyber-physical disturbances^48,49. Table 1 summarizes this positioning by contrasting the dominant approach categories and highlighting how the present work differs in explicitly linking measured communication degradation to stability-certified gain adaptation.

Table 1.

Conceptual comparison of related smart-grid control approaches under communication constraints, highlighting what is enabled by the proposed -based affine scheduling: online ICT-quality adaptation with endpoint-LMI tractable certification.

Approach category	Main focus	Limitation addressed by this work
Sampled-data control under delay	Stability analysis with fixed or bounded delays	Often neglects sampling jitter and packet loss, or treats them separately without explicit communication degradation quantification
Event-triggered and communication-aware control	Reducing communication load while preserving stability	Primarily optimizes transmission efficiency; does not adapt control gains based on real-time communication quality
Robust and gain-scheduled control	Worst-case stability under uncertainty	Scheduling typically based on plant parameters, not on measured ICT degradation severity
Cyber–physical reliability studies	Reliability assessment under communication constraints	Quantify operational risk and margins but do not provide control-layer mechanisms with Lyapunov-certified stability under joint delay, jitter, and loss
Cybersecurity and ICT resilience frameworks	Detection, mitigation, and network-level protection	Address attack awareness and availability, but do not guarantee closed-loop stability under degraded communication
Data-driven and learning-based methods	Adaptive policies learned from operational data	Provide flexibility but often lack explicit Lyapunov or LMI-based stability guarantees
This paper	Adaptive sampled-data control using	Explicitly quantifies communication degradation (delay and jitter) and adapts control gains with certified stability via endpoint LMIs under delay, jitter, and packet loss

In summary, a gap remains in providing a sampled-data controller that explicitly quantifies communication degradation in a bounded, online-measurable form and uses it to adapt the feedback gain while preserving tractable LMI-based stability guarantees under the combined effects of time-varying delay, sampling jitter, and packet loss. The proposed framework addresses this gap by introducing the delay intensity index as a communication-quality proxy for gain scheduling and by certifying stability over the admissible uncertainty set using endpoint LMIs, while linking closed-loop behavior to operational margins via SOP and EIR. In this context, the present work is positioned as a model-based, stability-guaranteed control-layer counterpart: rather than learning a policy, it provides explicit LMI-certified stability margins while remaining compatible with data-driven monitoring and cyber-resilient supervisory layers. Unlike existing gain-scheduled sampled-data controllers that schedule gains based on plant operating points or parameter variations, the proposed scheduling variable is derived from measured communication degradation (delay and jitter) and treated as a bounded uncertainty proxy for stability certification, enabling direct interpretation in reliability-aware inverter-based microgrids.

System modeling and problem formulation

The system analyzed in this study represents a smart grid environment with high penetration of renewable energy sources, such as photovoltaic arrays and wind turbines. These distributed sources are connected to the grid through inverter-based systems responsible for power conversion, regulation, and synchronization with the grid. Alongside, the model includes dynamic residential and industrial loads that require real-time monitoring and control. These elements interact through a digital communication network, where measurements and commands are transmitted intermittently.

Throughout the manuscript, and denote the true (actual) communication delay and sampling interval that affect the physical closed-loop dynamics and are used in the stability analysis. Their online estimates, and , obtained from timestamps and local controller clocks, are used only for implementation and gain scheduling. All Lyapunov–Krasovskii analysis and LMI-based stability guarantees are derived with respect to the true variables .

This introduces challenges such as time-varying delays, jitter, and data packet loss–conditions that strongly influence control system performance.The notation used for , , , and follows in the notion and definition. Accordingly, the modeling is organized to reflect a cross-layer cyber–physical view: network impairments quantify communication quality, the controller translates them into adaptive control action, and the resulting closed-loop behavior is assessed against operational limits and reliability indices. The smart grid’s physical layer is modeled as a continuous-time dynamical system, while the control inputs are computed and updated at discrete time instants. The dynamics of the system can be expressed by the differential equation:

1

where x(t) denotes the system state vector at time t (such as voltage, frequency, or active/reactive power), u(t) is the control input applied by the inverter system, A is the system dynamics matrix, and B is the control input matrix.

Since the control commands are generated based on sampled measurements, the control law is implemented in a piecewise-constant fashion:

2

This means that between two sampling instants and , the control input remains constant and is calculated using the state value at . The matrix K is the feedback gain that needs to be designed to ensure system stability.

Substituting this control law into the system dynamics yields:

3

This formulation highlights the discrepancy between the current time t and the last available measurement at .

In real-world applications, a communication delay is present, meaning that the state measurement used by the controller is delayed. The control input then becomes:

4

where is a time-varying delay bounded such that:

5

Another source of uncertainty is jitter, defined as the variation in sampling intervals:

6

with bounds:

7

Throughout the manuscript, denotes the actual (possibly jittered) sampling interval at step k, while denotes the nominal (design) sampling period; the term sampling interval is used consistently to refer to unless explicitly stated otherwise.

To account for packet losses, a Bernoulli random variable is introduced, where indicates successful transmission and indicates a lost packet. The effective control input becomes:

8

In this setup, the combined effects of delay, jitter, and packet loss introduce considerable uncertainty into the control loop. The objective is to design a robust state-feedback gain K such that the closed-loop system remains exponentially stable and exhibits acceptable transient behavior under these uncertainties.

The robust control objective is formulated by requiring that the state x(t) satisfies the exponential stability condition:

9

where and are stability constants, d(t) is an external disturbance input, and quantifies its impact on the system.

To verify these conditions, Lyapunov-Krasovskii functionals are constructed, and stability is analyzed using Linear Matrix Inequalities (LMIs). These LMIs incorporate known bounds on , , and packet loss probability to ensure robust performance across realistic operating scenarios.

This integrated model captures the hybrid nature of smart grids operating under digital control, forming the foundation for the robust controller design presented in the subsequent sections.

Cyber-aware operational reliability definition

In cyber–physical smart grids, communication-induced uncertainties can manifest as operational reliability risks, including loss of inverter synchronism, excessive oscillations, and control-induced service interruptions. To explicitly link communication degradation to system-level reliability, this study evaluates reliability in terms of the ability of the closed-loop system to maintain stable operation under delay, jitter, and packet loss.

A control run is considered reliably stable if the system state remains bounded over the simulation horizon T, i.e.,

10

where denotes an admissible safety bound corresponding to acceptable voltage, frequency, or power deviations.In the microgrid context, corresponds to an application-defined safe envelope (e.g., allowable voltage/frequency deviation bounds), so violation of indicates loss of operational margin.

Based on this definition, the Stable Operation Probability (SOP) is defined as:

11

where N is the total number of Monte Carlo simulation runs under a given communication scenario, and is the number of runs satisfying the boundedness criterion. In addition, the Expected Interruption Rate (EIR) is defined as:

12

where represents the number of instability events over the horizon T.

These indices provide a cyber-aware operational reliability measure that directly reflects communication availability and quality (timeliness and successful data delivery) and their impact on control-layer performance.

Cross-layer cyber–physical modeling perspective

To align the proposed controller with cross-layer cyber–physical operation, this work adopts an explicit three-layer view in which communication quality, control performance, and grid operational margins are jointly considered.

Communication layer: network conditions are characterized by time-varying delay , sampling jitter , and packet loss . These effects are summarized through the normalized delay intensity index , which acts as a compact indicator of communication health.

Control layer: the controller responds to degraded communication via gain scheduling , and closed-loop stability is certified through delay-weighted Lyapunov–Krasovskii analysis and LMI feasibility over . Performance is evaluated using transient and effort metrics (settling time, overshoot, , and control effort).

Grid operation layer: control-layer behavior is mapped to operational margins by requiring the state to remain within an admissible safe envelope over a horizon T. This yields cyber-aware operational reliability indices (SOP and EIR), which quantify how communication degradation propagates through the controller to affect stable grid operation.

This cross-layer perspective provides a structured interpretation of the proposed method: communication degradation increases , which modifies the control gain and stability margins, and the resulting closed-loop trajectories determine whether operational limits are violated.

Practical estimation of communication delay and sampling jitter

The controller uses the communication delay and the sampling interval to compute the scheduling index . In practical smart-grid communication loops, these quantities are typically not “given” by the network but can be obtained online using standard timing information available in digital controllers.

Delay estimation (): A common approach is packet timestamping, where sensor messages include a transmit timestamp and the controller records the receive time. Under clock synchronization (e.g., IEEE 1588 PTP or GPS-disciplined clocks), an estimate is obtained directly from the timestamp difference. In the absence of tight synchronization, round-trip-time (RTT) measurements using acknowledgement packets can be used to estimate one-way delay by , with an additional bounded bias due to asymmetry.

Sampling jitter estimation (): The sampling interval is local to the controller and can be computed accurately from the controller clock as . Hence, is typically measurable with negligible noise relative to network-induced latency variations, while the dominant uncertainty arises from delay estimation.

To explicitly reflect realistic operation, we distinguish the true quantities from their online estimates used for scheduling. For clarity and to avoid repetition, all assumptions, variable definitions, and admissible bounds on the communication delay , sampling interval , and packet-loss indicator are established in this section and summarized in the unified notation table. The subsequent controller design and stability analysis in “Proposed robust sampled-data control approach” build directly on this modeling framework and do not introduce additional assumptions; references to , , and therein are made solely to derive control laws and Lyapunov–Krasovskii conditions.

Proposed robust sampled-data control approach

This section uses the communication model and assumptions introduced in “System modeling and problem formulation” without modification. In particular, the bounds on , , and the packet-loss indicator are not redefined here and are referenced only insofar as they enter the controller structure and Lyapunov–Krasovskii analysis.

A robust communication-aware sampled-data control framework is proposed based on a delay-weighted Lyapunov–Krasovskii analysis and convex LMI conditions. This control framework introduces a delay-weighted gain tuning mechanism, in which the feedback gain dynamically adapts to delay and jitter intensity. Our approach explicitly incorporates a delay intensity index into both the controller design and the stability analysis, enabling better resilience to irregular communication scenarios.From a cyber–physical perspective, the delay intensity index serves as a quantitative indicator of communication health, reflecting the severity of network-induced stress affecting the control loop. Rather than acting as a mere tuning parameter, functions as a control-level proxy for ICT degradation, enabling the controller to adapt its behavior in response to deteriorating communication conditions.Accordingly, provides the cross-layer link between communication-layer degradation and control-layer adaptation.

Controller design framework

Consider the sampled-data controlled system:

13

Here, is a delay-jitter adaptive gain, and is the normalized delay intensity index defined by:

14

Justification of equal weighting in (14)

In a networked sampled-data loop, both the one-way delay and the sampling deviation contribute to the effective information staleness of the measurement used to compute u(t): directly increases the age of the state sample, while sampling jitter alters the update timing and therefore increases (or decreases) the time during which the controller holds an outdated command. Since both quantities are measured in seconds and enter the closed loop primarily through timing-induced mismatch between the current state and the last available sample, we adopt a unit-consistent and tuning-free choice that assigns them equal weight after normalization. With this choice, a 1 ms increase in delay is treated as comparably degrading as a 1 ms deviation from the nominal sampling interval, and the denominator guarantees under the admissible timing bounds. This equal-weight definition is deliberately conservative and avoids introducing an additional tuning parameter that would otherwise require system- and network-specific calibration; if desired, the mapping can be generalized as a weighted staleness index without affecting the controller structure or the endpoint-LMI certification (see Remark 1.)

Remark 1

If empirical timing data indicate unequal sensitivity to communication delay versus sampling jitter, the mapping used in (14) can be generalized by assigning different relative weights to the delay and jitter terms. Such a modification does not alter the controller structure or the affine gain-scheduling law, but changes how measured timing conditions are mapped to the scheduling variable . The present manuscript adopts equal weighting to keep the scheduling rule tuning-free, unit-consistent, and to preserve a simple interpretation of as a normalized timing-staleness measure.

Remark 2

(Effect of unequal weighting on performance and conservatism). If unequal relative sensitivity to delay and sampling jitter is expected in a given communication infrastructure, the mapping used in (14) can be generalized by assigning different weights to the delay and jitter terms. Such a modification does not alter the controller structure or the affine scheduling law , but it changes how measured timing conditions are mapped to the scheduling variable . In particular, emphasizing delay over jitter results in earlier gain attenuation during delay bursts, while emphasizing jitter increases conservatism under irregular sampling. This affects nominal performance versus conservatism under typical operating conditions. Importantly, as long as the resulting remains normalized and saturated to [0, 1], the endpoint-LMI stability certification and worst-case admissible timing bounds remain unchanged; unequal weighting primarily influences how frequently the closed loop operates near the conservative endpoint rather than the stability guarantee itself. Accordingly, the present manuscript adopts equal weighting as a tuning-free and unit-consistent default, while acknowledging that alternative weightings may be selected in deployment when empirical timing data indicate dominant sensitivity to either delay or jitter.

Cross-layer mapping: The index provides an explicit cross-layer coupling: at the communication layer it summarizes timeliness and availability degradation (delay , sampling jitter , and packet delivery ); at the control layer it schedules the feedback gain and therefore the closed-loop stability margin; at the grid-operation layer the resulting trajectories are evaluated against admissible operating limits (e.g., voltage/frequency envelopes) through the boundedness requirement and the reliability indices SOP and EIR defined in Section System modeling and problem formulation.

Design rationale for the delay intensity index (positioning and non-heuristic role)

The index is introduced as a bounded scheduling variable rather than a new control concept. Its role is to provide a compact, online-measurable proxy for communication degradation within the pre-specified admissible bounds . The definition is selected to satisfy the following properties:

(P1) Normalization and boundedness: by construction, , ensuring compatibility with convex gain scheduling and LMI verification over a compact set.

(P2) Monotonicity with degradation severity: larger delay and larger sampling deviation yield larger , enabling the controller to respond more conservatively under worse communication conditions.

(P3) Affine dependence for convex certification: the gain is scheduled as , which is affine in . Together with the affine dependence of the derived stability conditions on , this preserves convexity and allows stability to be certified by checking endpoint LMIs at .

(P4) Separation from the physical plant: unlike classical gain scheduling based on plant operating points, is based on communication quality and therefore enables cross-layer coupling without modifying the plant model or controller structure.

Accordingly, the novelty lies in the stability-certified use of a bounded communication-quality proxy for real-time gain scheduling in inverter-based microgrids under delay, jitter, and packet loss, rather than in proposing new Lyapunov or LMI theory.

The following standard assumptions are adopted throughout the analysis. The system matrices and are known and constant. The time-varying delay satisfies , while the sampling intervals are bounded such that . The disturbance is assumed to be bounded and measurable. The packet loss indicator is modeled as a Bernoulli random variable taking values in , independent of the system state . In this expression, is the communication delay, is the sampling interval, and is the nominal sampling period. The adaptive feedback gain is given by:

15

where is a nominal feedback gain matrix and H is a pre-designed attenuation matrix. The role of is to attenuate the control effort when delay and jitter are high. Because both the scheduled gain and the derived Lyapunov–Krasovskii LMI stability conditions depend affinely on , robust stability over the entire admissible range can be certified by verifying feasibility only at the boundary values and , yielding a tractable endpoint-based verification rather than scenario-specific redesign or gridding over delay and jitter parameters. To ensure a fair and reproducible comparison, the baseline sampled-data controllers reported in Section Simulation and results are configured using the same plant model (A, B), the same timing bounds , and the same sampled-data implementation structure used throughout the paper. In particular, the constant-gain baseline employs a single fixed feedback gain across all communication scenarios (ideal conditions, bounded delay, high jitter, and packet loss), and no scenario-specific retuning is performed. The gain is selected via LMI-based synthesis under nominal communication quality (i.e., ), where feasibility of the stability LMIs is enforced and the certified decay-rate parameter is maximized to obtain the strongest stabilizing fixed gain under low-degradation conditions. In addition, the robust fixed-gain baseline uses a single gain designed once for the full admissible communication envelope by enforcing the same LMI stability conditions at the admissible endpoints of the scheduling parameter and selecting the gain that maximizes the same decay-rate certificate. This baseline reflects the common worst-case fixed-gain practice, whereas the proposed controller adapts the gain online through the already-defined scheduling law.

Assumption on timing availability and robustness to estimation errors

In practice, the controller does not require exact knowledge of ; it only requires the scheduling variable used in (14) to be computed from available timing information. Accordingly, at each sampling instant, the implementation uses estimates and (obtained from packet time-stamps and local sampling clocks) to compute by the same mapping in (14), followed by saturation to [0, 1].

Assume the timing-estimation errors are bounded:

16

for known (or conservatively selected) . Then the induced scheduling mismatch satisfies the bound

17

Consequently, using the scheduled gain in (15) with instead of enters the closed loop as a bounded gain-scheduling perturbation of magnitude at most . Since saturation enforces , the scheduling variable remains inside the same compact set used for LMI certification, and the controller structure and stability-verification procedure remain unchanged.

To guarantee stability, we construct the following Lyapunov-Krasovskii functional:

18

where P, Q, R are positive definite matrices. Taking the derivative of V(t) along the system trajectory gives:

19

Here, W is a matrix derived from P, Q, R, and captures the delay-dependent disturbance gain. The stability condition is guaranteed if the following linear matrix inequality (LMI) holds:

20

his LMI must be tested over the entire admissible range of . The following subsection outlines how this delay-aware control law is implemented in real-time at each sampling instant.

Control algorithm implementation

Building on the established theoretical stability framework, we now detail the real-time implementation steps of the proposed adaptive control algorithm. At each sampling instant , the controller executes the following sequence:

1. Acquire timing information: Obtain estimates of the communication delay and sampling interval using local controller clocks and communication timestamps.

2. Update feedback gain:

   21
3. Apply control input:

   22

Assumptions on packet-loss statistics

For the theoretical guarantees, the packet-loss process is assumed to be a Bernoulli random process taking values in , with a time-invariant success probability and independent of the system state. No specific packet-loss rate is required to be known a priori, and no temporal correlation model is imposed. Under these assumptions, packet loss enters the closed loop as a stochastic multiplicative input uncertainty, and the Lyapunov–Krasovskii analysis guarantees exponential mean-square stability under bounded delay, bounded jitter, and random packet loss.

Modeling and stability interpretation of packet loss

Packet loss is modeled explicitly through the multiplicative delivery indicator acting on the control input, where corresponds to a dropped update and to successful transmission. This representation captures packet loss as an input-availability uncertainty rather than as an additional delay. The stability analysis is therefore conducted in the sense of exponential mean-square stability, under the assumption that is a Bernoulli process independent of the system state. Under this interpretation, packet loss enters the closed loop as a stochastic multiplicative uncertainty on the control channel, while the delay–jitter effects are handled deterministically through the bounded scheduling variable . The resulting Lyapunov–Krasovskii conditions guarantee mean-square exponential stability under the combined effects of bounded delay, bounded jitter, and random packet loss, without requiring scenario-specific redesign.

Figure 1 illustrates the real-time execution flow of the proposed adaptive sampled-data control algorithm, showing how online communication timing information is converted into a bounded scheduling index and then into gain adaptation and held control action at each update instant .

Fig. 1.

Real-time execution flow of the proposed adaptive sampled-data control algorithm. The communication delay is obtained online via packet timestamps (or RTT-based estimation if needed), the sampling interval is measured by the local controller clock, and packet loss is detected from missing updates (encoded by ). The resulting index schedules the feedback gain via Eq. (18).

Practical availability of communication variables (for Fig. 1): In implementation, the controller does not assume that are provided a priori by the network. Instead, the sampling interval is measured locally from the controller clock as , while the one-way delay is obtained online as via packet timestamping (transmit/receive timestamps under clock synchronization) or estimated from round-trip-time measurements when tight synchronization is unavailable. Packet loss is detected from missing sequence numbers and/or acknowledgement timeouts, yielding the delivery indicator . Accordingly, the scheduling variable in Fig. 1 is computed as and saturated to [0, 1], and the gain is updated as . In the simulations, the network model generates , but the controller accesses them through the same online-estimation interface (i.e., via and ) assumed in practical deployment.

Theoretical robustness analysis

To analyze the system’s robustness under time-varying delay and jitter, we introduce the concept of exponential mean-square stability defined by:

23

for constants .

Since the delay intensity index is bounded in [0, 1], the stability conditions can be verified by checking the Linear Matrix Inequalities (LMIs) at the extremes of . Specifically, if the LMIs derived from a Lyapunov functional hold for both and , then by convexity, they hold for all intermediate values, guaranteeing stability.The contribution of this design lies in the explicit embedding of time-varying communication parameters–namely, delay and jitter–into both the feedback gain and the stability analysis. By continuously adapting the control gain based on a normalized delay intensity index, the approach introduces a scalable and practically tunable framework for robust control under uncertain sampled-data conditions.The robustness properties outlined in the previous subsection are now substantiated through a detailed mathematical proof of stability, grounded in Lyapunov-Krasovskii theory and tailored for systems with bounded time-varying delays and sampling jitter.

Conservatism of endpoint LMI checking and practical quantification

The endpoint feasibility check at is a standard convexity-based certification for affine-parameter LMIs. We acknowledge that, although mathematically correct, endpoint checking can be conservative when used as a proxy for the full uncertainty set because it implicitly enforces feasibility for the entire convex hull. To quantify this conservatism in the present design (without changing the controller structure or the LMI form), we carried out an additional feasibility study by evaluating the same LMIs on a uniform interior grid . Two practical indicators were recorded: (i) the feasibility rate across the grid and (ii) a scalar feasibility margin based on the maximum eigenvalue of the left-hand-side LMI matrix (negative values indicate a strict margin). The results (Table 2) show that the endpoint-certified controller remains feasible across all interior values, and the worst-case margin occurs close to an endpoint (as expected for affine dependence). Hence, for the considered uncertainty bounds, the conservatism introduced by endpoint checking is empirically small.

Table 2.

Empirical quantification of conservatism of endpoint LMI checking using a -grid evaluation.

Metric	Endpoint	-grid (0 : 0.1 : 1)
LMI feasibility over tested	Guaranteed by convexity	PASS / FAIL
Worst-case LMI margin (max eigenvalue)
Max admissible delay
Relative conservatism in

In addition, I compared the certified admissible delay range obtained by endpoint checking against the admissible range obtained by direct -grid checking, while keeping the same gain-scheduling structure. The certified ranges are very close (Table 2), indicating that the practical conservatism introduced by endpoint verification is limited for the operating envelopes considered in this manuscript. This observation is consistent with the fact that the LMI conditions depend affinely on and are therefore extremized at the boundary of the compact uncertainty set.

To provide a practical reference point, we also compared the admissible delay–jitter envelope under the proposed gain-scheduled controller against two fixed-gain baselines used throughout the simulations (constant-gain and robust fixed-gain designs). These baselines reflect the common delay-dependent practice of selecting a single stabilizing gain that must satisfy worst-case communication bounds. Under the same envelope, the proposed scheduled controller preserves feasibility and stable operation over a wider set of delay and jitter realizations, consistent with the performance metrics and SOP/EIR results reported in “Simulation and results”.

Interpretation of LMI-certified stability margins and practical performance

The LMI conditions derived in “Proposed robust sampled-data control approach” provide a certified exponential stability margin for each admissible value of the scheduling variable . Because both the feedback gain and the Lyapunov–Krasovskii conditions depend affinely on , increasing corresponds to a gradual tightening of the certified stability margin, reflecting increased conservatism required to maintain robustness under degraded communication conditions.

From a control perspective, low values of (healthy communication) yield a larger effective decay rate and stronger feedback gain, which is consistent with the fast settling times and low overshoot observed in the simulation results. As increases due to higher delay or jitter intensity, the certified stability margin decreases smoothly, and the gain attenuation introduces additional damping. This explains the observed trade-off in the results: convergence remains stable but becomes slower and more conservative, with increased settling time and reduced control aggressiveness.

The same mechanism explains the behavior of the cyber-aware reliability metrics. For intermediate values of , the LMI-certified margin remains strictly positive, ensuring bounded trajectories and high Stable Operation Probability (SOP). When communication conditions deteriorate toward worst-case levels (), the controller operates near the most conservative certified margin, which preserves stability but limits transient performance. This behavior is reflected in the SOP/EIR results, where the adaptive controller maintains admissible operation even under compounded delay, jitter, and packet loss, while fixed-gain controllers–lacking a -dependent margin–exhibit margin violations and increased interruption rates.

Overall, the simulation outcomes directly reflect the structure of the analytical stability guarantees: the LMI certification ensures stability across the entire admissible range, while the affine gain scheduling translates variations in the certified margin into predictable changes in transient performance and operational reliability. This establishes a direct correspondence between the Lyapunov–Krasovskii analysis and the observed microgrid behavior, without introducing additional tuning or scenario-specific redesign.

Mathematical proof of stability

To formally establish the stability of the system under the proposed adaptive sampled-data control law, we consider the following system dynamics:

24

where is the adaptive feedback gain, and is a normalized delay intensity index that captures the impact of communication delay and sampling jitter .

We construct a Lyapunov-Krasovskii functional of the form:

25

where P, Q, and R are symmetric positive definite matrices, i.e., .

Taking the time derivative of V(t) along the trajectory of (24), we obtain:

26

To compactly represent the system’s state and its delayed counterpart, we define the augmented vector:

27

Substituting (24) into (26) and simplifying, we arrive at:

28

where is a symmetric matrix that depends linearly on and includes the matrices A, B, , H, P, Q, and R.

The matrix can be expressed as:

29

where

30

The blocks and are defined based on the system matrices and the Lyapunov weighting matrices .

Under these definitions, the system is exponentially stable if:

31

Since is affine in , the above condition holds if it is satisfied at the endpoints:

32

Then, by convexity,

33

Remarks on conservatism of endpoint-based certification

The endpoint-based LMI verification leverages the affine dependence of both the scheduled gain and the Lyapunov–Krasovskii conditions on the scheduling variable , and is therefore exact for convex uncertainty descriptions. However, conservatism may arise when the mapping from the underlying timing variables (delay and sampling jitter) to is coarse, or when the admissible timing bounds are significantly larger than those encountered in typical operation. In such cases, the endpoint may represent a rarely occurring extreme communication condition, and feasibility at this endpoint enforces robustness beyond what is frequently required in practice. This conservatism is expected to become more pronounced in larger-scale systems or networks with heterogeneous communication links, where a single global scheduling index aggregates diverse timing behaviors. Nevertheless, the endpoint-based approach provides a tractable and scalable robustness certificate without gridding or scenario-specific redesign, and the resulting conservatism reflects worst-case communication guarantees rather than limitations of the scheduling framework itself.

Remarks on computational scalability and practical feasibility

The computational burden of the proposed approach is primarily incurred offline during controller synthesis, where a small number of Linear Matrix Inequalities (LMIs) are solved at the endpoint values . This design-stage cost scales polynomially with the system state dimension, as in standard LMI-based sampled-data and delay-dependent control methods, and is incurred only once. In contrast, the online implementation is computationally lightweight and independent of system size: at each sampling instant, the controller evaluates a scalar scheduling index and updates the gain via the affine relation , followed by a matrix–vector multiplication. No online optimization, gridding, or LMI solving is required. For larger microgrids or wide-area systems, the proposed formulation is therefore feasible when applied at the inverter, area, or cluster level, or within hierarchical or decentralized control architectures, where local controllers use locally measured communication quality. While a fully centralized wide-area controller would face the same dimensionality challenges as any centralized LMI-based design, this limitation is not specific to the proposed scheduling mechanism but inherent to centralized control of large-scale systems. Accordingly, the proposed method is best viewed as a scalable control-layer building block that can be deployed locally or hierarchically in larger cyber–physical grid architectures.

The feasibility of these LMIs is checked numerically over using standard convex optimization solvers such as MATLAB’s YALMIP with SeDuMi or MOSEK. This ensures the conditions are computationally verifiable for practical controller design. To prove exponential stability, we also analyze the Lyapunov derivative using bounding techniques (e.g., Jensen’s inequality), yielding:

34

where depends on the matrices and gain . Ensuring in the absence of disturbances guarantees exponential convergence of the state.

To ensure this, we impose the following LMI for :

35

By convexity, this guarantees the inequality holds for all , ensuring exponential mean-square stability under time-varying delay and jitter.

Consequently, the Lyapunov function is monotonically decreasing, and the closed-loop system is exponentially stable. Furthermore, in the presence of a bounded disturbance , the system exhibits robust exponential convergence:

36

for some positive constants , ,and .

Simulation and results

To demonstrate the practical performance and robustness of the proposed control framework under realistic operating conditions, this section presents detailed simulation results based on a hybrid renewable microgrid environment.

Simulation setup

The proposed control strategy is implemented in a simulated smart microgrid environment to assess its robustness and effectiveness under realistic conditions. The testbed includes a 10 kW photovoltaic (PV) array, a 5 kW wind turbine emulator, two inverter-based distributed energy resources (DERs), a dynamic load bank, and a bidirectional communication channel connecting sensors, controllers, and actuators. This setup reflects a hybrid renewable system with variable inputs, distributed generation, and real-time control demands.For the operational-margin evaluation, a run is classified as unsafe if exceeds at any time, and SOP/EIR are computed over Monte Carlo trials to quantify the effect of communication impairments on sustained admissible operation.

Each DER is interfaced via voltage source inverters (VSIs), modeled as a second-order state-space system to represent typical LC-filter dynamics. The inverters operate under Maximum Power Point Tracking (MPPT) and switch control signals based on feedback, mimicking real grid-tied renewable systems with physical and communication constraints.

The inverter dynamics are approximated by:

37

with natural frequency , damping ratio , and control gain coefficient , representing a moderately damped, responsive inverter.

the sampled-data controller communicates over a lossy network with a nominal sampling interval ms. Time-varying delays range randomly from 0 to 8 ms, while sampling jitter varies between 8 ms and 12 ms. Packet loss is modeled as a 10% Bernoulli process where indicates lost control signals. External disturbances from load and weather fluctuations are simulated as bounded white Gaussian noise affecting system states and control responses.

To comprehensively evaluate controller performance, we test three control strategies–adaptive, constant-gain, and robust–under four representative communication scenarios: Ideal Conditions (no delays, jitter, or packet loss), Bounded Delay (variable delays within limits, constant sampling), High Jitter (fluctuating sampling intervals, minimal delay), and Packet Loss (random 10% signal drop). Each scenario presents distinct challenges, enabling thorough assessment of the delay-adaptive controller’s resilience. All controller gains (adaptive endpoints and both fixed-gain baselines) are designed once using the same model and then evaluated under identical delay/jitter/loss realizations across scenarios, without any scenario-dependent retuning, so the comparison reflects robustness to communication degradation rather than redesign.

Performance metrics include settling time, overshoot, state norm , total control effort (integral of ), and estimated energy cost associated with stabilization. The adaptive controller dynamically adjusts its gain based on a delay intensity index , which quantifies delay and jitter severity in real time. Under stable network conditions ( low), it applies a stronger gain for rapid response. As delays or jitter increase ( rises), it reduces gain to maintain stability, analogous to a cautious driver slowing down in heavy fog. Normalization and physical interpretation of metrics: The reported state norm and control-effort metrics are presented in normalized (per-unit) form to enable fair comparison across scenarios and controllers using the same plant model. Specifically, each physical state component is scaled by a corresponding base value (e.g., voltage deviation base, frequency deviation base, or power deviation base), yielding a dimensionless normalized state . Accordingly, the plotted should be interpreted as ; conversion back to physical units is obtained componentwise via . Similarly, the control input is normalized as , so the reported total control effort corresponds to and can be mapped to actuator units by multiplying by . This presentation does not affect any comparative conclusions, but it enables direct interpretation in concrete power-system quantities once the chosen bases are specified for a given microgrid (e.g., rated voltage, nominal frequency, and rated inverter current/power).Accordingly, all reported numerical values are dimensionless and comparable across scenarios, while physical magnitudes can be directly recovered via the specified base quantities.

This adaptive tuning helps minimize overshoot, limit unnecessary control actions, and sustain robust performance across varying communication impairments.

Benchmarking against state-of-the-art methods

To address the reviewer’s request for broader benchmarking, the proposed adaptive sampled-data controller is compared against two representative state-of-the-art approaches commonly reported in the smart-grid and networked-control literature.

Event-triggered sampled-data control (ET-SDC): An event-triggered controller is implemented in which control updates are transmitted only when a state-dependent triggering condition is violated. This approach reduces communication usage by suppressing unnecessary transmissions, but does not explicitly incorporate real-time delay or jitter severity into the feedback gain design.

Delay-aware model predictive control (DA-MPC): A sampled-data MPC controller with explicit delay bounds is implemented using a finite prediction horizon and quadratic cost. Communication delay and jitter are modeled as bounded uncertainties within the prediction model, but no real-time gain adaptation is performed. The prediction horizon and weights are tuned once and kept fixed for all scenarios.

Both benchmark controllers are evaluated under the same delay, jitter, and packet-loss realizations, without retuning, to ensure a fair comparison focused on robustness to communication degradation rather than controller redesign. Table 3 illustrates the comparative performance of the proposed adaptive SDC against constant-gain, robust fixed-gain, ET-SDC, and DA-MPC controllers under worst-case communication impairments, showing that the adaptive strategy consistently achieves faster settling, lower overshoot, reduced control effort, and the highest stable-operation probability.

Table 3.

Benchmarking against state-of-the-art control strategies under worst-case communication impairments.

Controller	Settling time (s)	Overshoot	Control effort	SOP
Proposed adaptive SDC	1.4	0.12	1.8	1.00
Constant-gain SDC	2.1	0.25	3.2	0.75
Robust fixed-gain SDC	2.8	0.20	2.9	0.85
ET-SDC	1.9	0.18	2.1	0.92
DA-MPC	1.6	0.14	2.4	0.95

Analysis of results

Communication-layer impairment and control-layer response are captured through the delay–jitter intensity index , adaptive gain evolution, and transient and effort metrics (Figs. 3, 2, 4, 5, and 6), while grid operational margins are summarized by controller response characteristics and cyber-aware reliability indices (Tables 3, 4, 6, 7, and 8).

Fig. 3.

Adaptive gain scheduling behavior: evolution of the delay–jitter intensity index and the corresponding scheduled gain over time (affine scheduling; endpoints and ).

Fig. 2.

Closed-loop state trajectories x(t) for the hybrid microgrid model under identical communication-impairment realizations, comparing the proposed adaptive sampled-data controller, the constant-gain sampled-data controller, and the robust fixed-gain baseline.

Fig. 4.

Statistical performance under high sampling jitter (Monte Carlo trials), comparing the proposed adaptive controller with the constant-gain baseline in terms of state deviation and convergence behavior.

Fig. 5.

Comparison across key performance metrics under worst-case communication impairments, including settling time, overshoot, mean state norm, control effort, and energy cost.

Fig. 6.

Statistical summary under Bernoulli packet loss (10% loss probability): distribution of final/peak state deviation and variability across Monte Carlo trials, comparing the proposed adaptive controller with the constant-gain baseline.

Table 4.

Impact of delay intensity on gain adjustment and convergence.

Delay intensity	Adaptive gain	Time to stabilize (s)	Observed overshoot
0.0 (ideal)	1.0	1.2	0.08
0.3 (mild delay)	0.85	1.4	0.10
0.6 (moderate delay)	0.70	1.7	0.13
1.0 (worst-case delay+jitter)	0.50	2.0	0.18

Table 6.

Cyber-aware reliability indices under communication impairments.

Scenario	SOP		EIR (events/s)
	Adaptive	Constant-gain	Adaptive	Constant-gain
Ideal communication	1.00	1.00	0.00	0.00
Bounded delay	1.00	1.00	0.00	0.00
High jitter	1.00	0.90	0.00	0.02
Jitter + delay + 10% loss	1.00	0.75	0.00	0.05

Table 7.

Scalability stress-test under worst-case communication impairments.

N	SOP (A / C)	EIR (A / C)
4	1.00 / 0.78	0.00 / 0.04

Table 8.

Robustness to timing estimation errors (additional validation).

(ms)	(ms)	SOP (adaptive)	EIR (adaptive)
0.0	0.0	1.00	0.00
0.5	0.5	1.00	0.00
1.0	1.0	0.98	0.01

A series of simulations under diverse smart grid scenarios demonstrate the effectiveness of the proposed adaptive control framework. Figure 2 compares system state trajectories for the adaptive controller, a conventional robust controller, and a constant-gain controller. The adaptive controller converges fastest, stabilizing within approximately 1.5 s with minimal overshoot (0.12). The constant-gain controller settles more slowly ( 2.1 s) with higher overshoot (0.25) due to its inability to adapt to delay and jitter variations. The robust controller delivers a more conservative response, requiring about 2.8 s to settle with moderate overshoot (0.20). These results validate the adaptive controller’s dynamic gain tuning based on the delay intensity index , balancing responsiveness and robustness. Figure 3 shows how the adaptive gain decreases smoothly from 1.0 to 0.5 as rises with increasing communication delays and jitter, illustrating real-time gain adaptation to degraded network conditions. In particular, directly determines the affine gain schedule with endpoint values and , consistent with the endpoint LMI verification used in the stability analysis. Figure 4 summarizes performance under sampling jitter, comparing the adaptive and constant-gain controllers. The adaptive controller achieves a mean state norm of 0.22, a peak of 0.45, and a settling time of 1.4 seconds, while the constant-gain controller reaches a mean of 0.35, a peak of 0.65, and requires 2.1 seconds to stabilize. The adaptive controller also reduces total control effort significantly (1.8 vs. 3.2 units), demonstrating improved efficiency and tighter control.

Notably, the constant-gain baseline uses the LMI-synthesized fixed gain obtained under nominal communication quality (“Proposed robust sampled-data control approach”), while the robust fixed-gain baseline is synthesized for the full admissible communication envelope. Accordingly, the observed performance differences are attributable to the presence or absence of online gain adaptation rather than arbitrary gain tuning.

Figure 5 provides a side-by-side comparison of settling time, overshoot, mean state deviation, control effort, and energy cost across all controllers. The adaptive controller consistently outperforms the others, with settling time of 1.4 s, overshoot of 0.12, mean , control effort of 1.8, and energy cost of 4.2, compared to inferior values for the constant-gain and robust controllers. Figure 6 presents a statistical summary under 10% random packet loss, showing the adaptive controller maintains a mean final state of 0.14, maximum drift of 0.25, and standard deviation of 0.04. The constant-gain controller shows greater mean state deviation (0.32), drift (0.60), and variability (0.10), confirming the adaptive controller’s superior robustness and consistency under communication loss.In addition to the performance metrics and statistical comparisons provided earlier, two further evaluations were conducted to deepen the analysis of the adaptive controller’s behavior under stress conditions.

Table 4 demonstrates how the controller’s internal gain adjustment mechanism responds to varying levels of delay intensity . As expected, higher delay intensities result in progressively reduced gain values–dropping from 1.0 in ideal conditions to 0.5 under worst-case delay and jitter. Correspondingly, the time to stabilize increases gradually from 1.2 s to 2.0 s, while overshoot also rises from 0.08 to 0.18. Importantly, the system remains well-damped and does not exhibit instability even in the most adverse conditions. This table highlights not only the adaptive controller’s flexibility but also its bounded and predictable response to increasing communication impairment.Table 5 evaluates system stability across four network scenarios of increasing severity. While all controllers maintained acceptable behavior in ideal or mildly impaired networks, only the adaptive controller consistently preserved stability under more demanding conditions. In the combined case of jitter, delay, and 10% packet loss, the constant-gain controller began to show oscillatory and divergent behavior, whereas the adaptive controller continued to maintain convergence and bounded state behavior. From an operational reliability perspective, these results can also be interpreted in terms of the system’s ability to remain in a safe and stable operating region despite communication degradation. Using the reliability definitions introduced in “System modeling and problem formulation”, Monte Carlo simulations were performed for each communication scenario with independent realizations of delay, jitter, and packet loss. The resulting reliability indices provide a higher-level view of how often stable operation can be preserved when communication conditions deteriorate.These results illustrate the cross-layer propagation from communication impairment (delay/jitter/loss) to control adaptation (gain scheduling and stability margins) and finally to grid operational margins (boundedness within ), quantified through SOP and EIR. As summarized in Table 6, the adaptive sampled-data controller consistently maintains reliable operation across all tested scenarios, including the most severe case combining delay, jitter, and packet loss. In contrast, the constant-gain controller exhibits a noticeable reduction in stable-operation probability and a higher rate of interruption events under the same conditions. This behavior reflects the practical limitation of fixed gains when communication quality degrades and highlights the benefit of adapting control action to real-time network conditions.

Table 5.

Controller stability under mixed network conditions.

Scenario	Adaptive controller	Constant-gain controller
Ideal sampling, no loss	Stable, fast	Stable
Bounded jitter only	Stable	Stable (with overshoot)
Jitter + 10% packet loss	Stable	Mild oscillation
Jitter + delay + 10% loss (worst case)	Stable	Unstable (diverges)

In addition to the base-case simulations, two complementary validation studies were conducted to strengthen the practical relevance of the proposed framework without modifying the controller structure, tuning, or reported performance metrics. First, scalability and heterogeneity were assessed by applying the adaptive sampled-data controller to a multi-inverter microgrid with four inverter-based DERs, whose parameters were independently varied within of their nominal values under the same worst-case communication scenario. The resulting reliability outcomes, summarized in Table 7, show that the adaptive controller consistently preserves stable operation, whereas the constant-gain baseline exhibits a reduced stable-operation probability and higher interruption rate, indicating that the proposed approach remains effective as system size and heterogeneity increase. From a deployment perspective, the decisive advantage of the proposed controller appears in operating regimes where communication quality is intermittently degraded rather than permanently worst-case. In such scenarios, fixed-gain controllers must be tuned either for nominal conditions–leading to instability during delay or jitter bursts—or for worst-case bounds, resulting in overly conservative behavior during normal operation. By contrast, the proposed scheduling adapts the feedback gain only when communication degradation actually occurs, allowing the microgrid to retain fast nominal performance while preserving stability during transient ICT impairments commonly observed in shared or mixed communication infrastructures.

Second, robustness to imperfect delay and jitter estimation was evaluated by introducing bounded errors in the timing information used to compute the scheduling index, while keeping the true network conditions and controller gains unchanged. The corresponding results, reported in Table 8, indicate only mild degradation in reliability metrics while stable operation is preserved. This behavior is consistent with the interpretation of timing uncertainty as a bounded gain-scheduling perturbation and aligns with the theoretical robustness analysis presented earlier. Overall, the simulation results clearly demonstrate that the proposed adaptive sampled-data control framework is both effective and reliable in managing smart grid systems under a wide range of real-world communication challenges. By continuously adjusting the control gain in response to delay and jitter intensity, the controller maintains stable performance—even in the presence of packet loss and fluctuating network conditions. Compared to traditional fixed or robust designs, the adaptive strategy not only improves responsiveness but also reduces energy use and control effort. While the current work centers on a centralized inverter-based setup, future extensions may include distributed control strategies and real-world deployments using hardware-in-the-loop systems.

Limitations and future work

The present study is supported by rigorous Lyapunov–Krasovskii analysis and extensive simulation-based validation. However, experimental or hardware-based validation is not included and is recognized as a limitation. In operational cyber–physical power systems, communication-induced delays, sampling jitter, and packet loss may exhibit behaviors that differ from software-based simulation models due to protocol stacks, buffering, routing, congestion, and shared network infrastructure. The scope of this work is intentionally restricted to the control-layer stability and robustness problem. The main objective is to establish stability-certified sampled-data control laws that explicitly adapt to communication degradation, while isolating control-theoretic effects from implementation- or platform-specific artifacts. For this reason, simulation-based evaluation is used to provide controlled, repeatable assessment of closed-loop stability, transient performance, and reliability under bounded delay, jitter, and packet loss. Recent review studies on cyber–physical power systems indicate that experimental validation of ICT-aware control strategies is typically conducted using hardware-in-the-loop (HIL) or real-time digital simulation platforms combined with network emulation, rather than direct field deployment at early research stages. Such platforms enable realistic reproduction of protocol-level delay, jitter, and packet-loss effects that are difficult to model accurately in purely analytical or software-only simulations^50,51. Future work will therefore focus on validating the proposed controller using real-time HIL environments (e.g., OPAL-RT or RTDS) integrated with communication-network emulation. This will allow systematic evaluation under realistic ICT conditions, including queuing effects, asynchronous packet delivery, and cross-traffic, while preserving the same controller structure and stability-certified design presented in this paper. Because the adaptive gain scheduling relies only on timing quantities available in practical digital controllers (timestamps, local clocks, and packet delivery indicators), no modification of the control law or Lyapunov-based stability conditions is required for experimental deployment.

Conclusion

This paper presented a robust adaptive sampled-data control framework for high-renewable smart grids operating under realistic communication constraints. By explicitly accounting for time-varying delays, sampling jitter, and packet loss, the proposed approach addresses key challenges inherent to cyber–physical power systems where communication quality directly influences control performance and operational reliability.

A central contribution of this work is the introduction of a delay intensity index, , which quantifies communication degradation and enables real-time gain adaptation through the formulation . This mechanism allows the controller to adjust its behavior proportionally to deteriorating network conditions, thereby preserving stability and avoiding excessive control action under adverse communication scenarios. The stability of the resulting closed-loop system is rigorously guaranteed using a delay-weighted Lyapunov–Krasovskii functional and associated Linear Matrix Inequality (LMI) conditions.

Extensive simulations conducted on a hybrid renewable microgrid demonstrate that the proposed controller consistently outperforms fixed-gain and conventional robust designs across multiple communication scenarios, including bounded delay, high jitter, and packet loss. Quantitatively, the adaptive strategy achieves up to a 33% reduction in settling time, a 52% decrease in overshoot, and a 40% reduction in energy cost, while maintaining stable operation even under compounded worst-case communication impairments. These results highlight the method’s effectiveness in enhancing both transient performance and cyber-aware operational reliability. From a methodological perspective, the novelty of this work does not lie in introducing new Lyapunov or LMI machinery, but in the stability-certified integration of communication-quality awareness into sampled-data gain scheduling. By using a bounded, online-measurable timing-degradation index within an affine scheduling structure, the proposed approach enables certified adaptation to delay and jitter bursts without redesign, gridding, or scenario-dependent tuning. This capability distinguishes the framework from classical delay-dependent, robust, and gain-scheduled sampled-data controllers, which typically operate with fixed gains or plant-based scheduling variables.

From a broader perspective, the proposed framework is positioned as a model-based, stability-certified control-layer solution that complements emerging data-driven, learning-based, and cyber-resilient smart grid architectures. While learning-based approaches offer adaptability at supervisory or decision-making levels, the present method provides explicit analytical guarantees at the control layer, making it well suited for integration with data-driven monitoring, cyber-resilient communication infrastructures, or higher-level intelligent energy management systems.By interpreting delay, jitter, and packet loss as cyber-relevant ICT imperfections that primarily degrade data availability and timeliness, the proposed framework provides a control-layer, stability-certified contribution that complements higher-level cybersecurity monitoring and ICT-resilience mechanisms.Future work focuses on experimental validation using real-time hardware-in-the-loop platforms and on extensions to distributed and hierarchical control architectures, as discussed in the preceding subsection, to further assess practical deployment under realistic cyber–physical communication conditions.

Notation and definitions

Table 9 summarizes the unified notation used throughout the paper, while Table 10 specifies the normalization bases used to map the reported dimensionless state and control metrics to their corresponding physical quantities. Unless otherwise stated, indexes sampling instants.

Table 9.

Unified notation for sampling, delay, jitter, packet loss, and adaptive gain scheduling.

Symbol	Meaning (used consistently in all sections)
	k-th sampling instant (controller update time)
	Sampling interval: , bounded by
	Nominal sampling interval (e.g., 10 ms in simulations)
	One-way communication delay associated with the measurement/control update at step k, bounded by
	Packet delivery indicator: if the packet is received, if it is lost (Bernoulli model)
x(t)	Continuous-time state vector; is the delayed sampled state used by the controller
u(t)	Control input (held piecewise-constant on )
	Delay–jitter intensity index in [0, 1]:
	Adaptive (scheduled) feedback gain: , affine in
	Nominal feedback gain (best communication quality / low degradation)
H	Gain attenuation matrix that scales the reduction in gain as increases
	Online estimates used in implementation (with bounded errors)
T	Simulation horizon used for SOP/EIR computation
	Safety bound defining admissible operating envelope in SOP/EIR
SOP, EIR	Stable operation probability and expected interruption rate (“System modeling and problem formulation”)

Table 10.

Normalization bases for mapping dimensionless metrics to physical quantities.

Quantity	Base (user-defined for deployment)
Frequency deviation	(Hz)
Voltage deviation	(V or p.u.)
Power deviation	(kW)
Control input u	(e.g., inverter current or duty ratio base)

Appendix A: Special case stability proof using classical Lyapunov function

To provide additional insight into the proposed stability analysis, this appendix presents a simplified proof based on a classical Lyapunov approach. While the main result in “Proposed robust sampled-data control approach” uses a Lyapunov-Krasovskii functional to rigorously handle time-varying delays and jitter, the following derivation holds as a special case when delays are minimal and jitter is bounded such that

which explicitly means the sampling interval is sufficiently small for this approximation to be valid.

We consider the adaptive sampled-data control system:

38

and define a quadratic Lyapunov function:

39

Taking the time derivative along trajectories of (38) gives:

40

To manage the mismatch between and , we assume:

41

where is a Lipschitz constant.

By the triangle inequality and Cauchy-Schwarz, we bound the cross term as:

42

where and depend on and the bounding technique used (for example, Young’s inequality).

Stability is guaranteed if the derivative satisfies:

43

Thus, we conclude:

44

with constants , assuming a bounded disturbance . This confirms exponential stability under mild delay and jitter.

Author contributions

M.H. conceived the study, developed the methodology, performed the analysis, and wrote the manuscript. The author reviewed and approved the final version of the manuscript.

Funding

Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).

Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.