With the increasing incorporation of Renewable Energy Sources (RESs) like solar Photovoltaic (PV) systems, maintaining frequency stability has turned out to be a significant challenge owing to decreased system inertia. Despite numerous developments in Load Frequency Control (LFC), existing solutions largely overlooked the issue of Under Frequency Load Shedding (UFLS) relay failure during rapid frequency decline, which led to widespread blackouts. To address this critical gap, a novel intelligent control framework integrating the Fuzzy Doubleton Parabolic Inference System (FDPIS) and the Proportional Quad-Alpine Integral Derivative (PQAID) controller for UFLS relay failure mitigation and enhanced LFC in Solar-PV systems is proposed. Primarily, the Direct Current (DC) power from the solar module is fed into the DC-DC boost converter and Maximum Power Smoothstep Point Tracking (MPSPT) algorithm. A capacitor bank failure is detected using FDPIS, and voltage stabilization is ensured through a Savitzky-Golay Dynamic Polynomial-Z Voltage Restorer (SGDP-ZVR). To predict electrical load demand accurately, a hybrid Deep Learning (DL) model, Deep Dualplus Softshrink Pan–Long Short Term Inverse Parzen Memory (2DSP-LSTIPM), is employed, delivering a high accuracy of 98.98 % with a Root Mean Squared Error (RMSE) of 0.002. When demand exceeds thresholds, transmission overload is mitigated using an Inductive Snubber Cubic Circuits–STATCOM (IS2C-STATCOM). The frequency deviation is identified via FDPIS, followed by the Rate Of Change Of Frequency (ROCOF) analysis. If a UFLS relay failure is detected, then the PQAID controller is activated to ensure stable operation. The proposed PQAID achieves a peak time of 1.91 ms, significantly outperforming traditional PID, PI, and PD controllers in transient and overshoot metrics. Simulation results on the HEDGW dataset assess the proposed approach’s robustness and low time complexity. The system demonstrates superior relay fault detection (fuzzification/defuzzification times of 452ms/463ms) and faster rule generation (597 ms) compared to conventional fuzzy systems. Overall, the proposed methodology provides a comprehensive, real-time, and scalable solution for enhancing frequency stability, relay fault mitigation, and load management in solar PV-based smart grids. • Integrates FDPIS and PQAID for real-time UFLS relay failure detection and mitigation in solar PV systems. • Proposes novel 2DSP-LSTIPM deep learning model achieving 98.98 % demand prediction accuracy with RMSE of 0.002. • Introduces SGDP-ZVR for voltage stabilization during capacitor bank faults using Savitzky-Golay filtering. • Deploys IS2C-STATCOM for efficient transmission overload control with fast reactive power regulation. • Enables seamless SCADA/EMS integration via OPC-UA protocol for smart grid compatibility and deployment.

Parkinson's disease patients (PD) display changes in the levels of circulating peripheral immune cells. A declining lymphocyte count and a raised neutrophil-to-lymphocyte ratio (NLR) may result from an impaired adaptive response and increased levels of inflammation. This study sought to identify circulating peripheral immune cell changes in PD patients. PD and healthy controls (HC) were recruited from the East London Parkinson's Disease (ELPD) project. Numbers of immune cells from peripheral blood samples were cross-sectionally analysed across three (pre-, within 6months of PD diagnosis, and post-PD diagnosis) or two (pre- and post-PD diagnosis) time points and compared with HC. The study included 145 (85 male (M), 60 female (F)) PD and 73 (47M, 26F) HC. The two-time point analysis included all participants, whereas the three-time point analysis included 41 (25M, 16 F) PD and 55 (36M, 19 F) HC. Lymphocyte count and NLR were not significantly different between PD and HC. There was a non-significant trend towards lower lymphocyte count in PD following diagnosis in an unadjusted model for both the two-time point (OR=0.76; CI=0.56-1.02, p=0.069) and three-time point (OR=0.62; CI=0.37-1.30, p=0.064) analyses, but in both analyses this trend disappeared when adjusting for confounding factors. Levodopa equivalent daily dose was not significantly associated with lymphocyte count or NLR. Our study revealed no significant difference in lymphocyte count or NLR between PD and HC. There was a non-significant trend towards lower lymphocyte count following diagnosis in PD.

Abstract Odd–harmonic repetitive control (OHRC) is a control scheme specifically designed for reference tracking and disturbance rejection of odd–harmonic signals, which are commonly found in servo systems and various power applications. However, OHRC suffers from critical limitations in rejecting multi-periodic disturbances, handling aperiodic signals, and maintaining stability under plant uncertainties. These drawbacks restrict its applicability in real-world systems with complex disturbances and time-varying dynamics. To overcome these challenges, this paper proposes ELM–OHRC, a control framework that integrates OHRC, a PD stabilizer, and an extreme learning machine (ELM) module. In this framework, OHRC is responsible for accurately tracking periodic signals composed of odd–harmonic components and rejecting corresponding disturbances, while the PD controller ensures closed-loop stability of the plant. Meanwhile, the ELM serves as a nonlinear adaptive estimator that complements the model-based OHRC, enabling effective rejection of complex disturbances beyond the scope of odd–harmonic components. Stability of the closed-loop system is ensured through Lyapunov–based analysis. Simulation results show that the proposed ELM–OHRC provides improved tracking accuracy and enhanced steady-state disturbance rejection under aperiodic and unmodeled periodic disturbances, compared with conventional OHRC and switching-based hybrid approaches. The learning-based disturbance estimation enables effective compensation of complex and unmodeled disturbances while preserving robust closed-loop performance.

The present work presents the dynamic modeling and nonlinear control of a 5-DOF artificial human arm, which can efficiently replicate human upper limb movements. A control-oriented 5-DOF rigid-body dynamic model was formulated by employing the Euler-Lagrange equation, which considers inertia, centrifugal, and gravitational forces. A total of four control approaches, namely Proportional-Derivative Control (PD), Computed Torque Control (CTC), Linear Quadratic Regulator (LQR), and Adaptive Sliding Mode Control (ASMC), were proposed and analyzed for their trajectory tracking abilities. Lyapunov stability theory was used to validate the convergence of the ASMC and CTC control approaches, whereas the optimal control gains were calculated for the LQR control approach by linearizing the state-space model. The simulation outcomes demonstrated that the ASMC control approach provides the highest level of trajectory tracking accuracy with an RMS error of 0.012 rad, compared to other control approaches such as CTC (0.041 rad), LQR (0.056 rad), and PD control (0.084 rad). The ASMC control strategy was found to have the fastest settling time of 0.78 s and the smallest overshoot of 2.1%, which confirms its robustness against parametric uncertainties. However, it is achieved at the expense of an increased control effort, with an average torque norm of 15.5 Nm, whereas the PD method requires the least control energy (6.2 Nm) but has poor accuracy. These results have demonstrated the effectiveness of nonlinear robust control, especially ASMC, in improving the tracking capability of bio-inspired robotic manipulators.

Lower-limb exoskeleton robots play a pivotal role in rehabilitation medicine and assistive augmentation, where precise dynamic modelling and trajectory tracking control are fundamental to effective assistance. Existing models predominantly focus on hip and knee rotational degrees of freedom, with insufficient attention to ankle dynamics and pelvic translation. To address these limitations, this paper establishes a sagittal-plane dynamic model comprising nine generalised coordinates, treating the human lower limb and exoskeleton as an integrated coupled system. A seven-segment kinematic model encompassing the trunk, bilateral thighs, shanks, and feet is constructed via a modified Denavit-Hartenberg parameter method, and dynamic equations are derived using Lagrangian formulation. Three control strategies-PD control, PD with gravity compensation, and the computed torque method-are designed and evaluated through simulations using gait data from five subjects (two self-collected, three from a public dataset) acquired via Vicon motion capture. Results demonstrate that the computed torque method achieves a joint angle tracking root mean square error (RMSE) of 0.59°, representing an 86.3% improvement over conventional PD control, while maintaining a low control torque RMS of 4.44 N·m. The controller exhibits stable tracking performance across walking speeds of 0.4-1.45 m/s, validating the effectiveness of the proposed model and control strategies.

BackgroundParkinson’s disease (PD) is a progressive neurodegenerative disorder that impairs motor function, thereby influencing daily activities, including sit-to-stand (STS) motion. Dopaminergic medication improves motor symptoms; however, its effects on neuromuscular control during STS motion remain unclear. This study investigated the effects of dopaminergic medication on muscle synergy and kinematic performance during STS motion in patients with PD.MethodsFourteen patients with PD performed STS motion in the OFF and ON medication states. Surface EMG data from eight trunk and lower limb muscles and kinematic data of the center of mass (COM) trajectory were recorded. Muscle synergies were extracted using non-negative matrix factorization to assess temporal features and activation patterns. Kinematic features, including STS duration, time to seat-off, and COM displacement angle (initial to seat-off), were analyzed.ResultsDopaminergic medication significantly improved muscle synergy, achieving earlier initiation of the seat-off synergy and improved coordination between the propulsive and postural stabilization synergies. Neuromuscular improvement showed associations with changes in functional performance. Kinematic analysis revealed that the ON state was marked by shorter movement duration, reduced seat-off time, and a downward COM trajectory. These findings indicated that dopaminergic medication improves muscle synergy activation timing to enhance movement efficiency.ConclusionThese findings suggest that dopaminergic medication enhances the temporal precision of neuromuscular coordination and resolves the dysfunctional compensatory strategies during STS motion. These results provide novel insights into how dopamine modulates motor control in PD, with implications for clinical assessment and rehabilitation.

This paper addresses the speed control problem of a DC motor in the presence of nonlinearities, disturbances, and unmodeled dynamics by proposing a neural backstepping control scheme based on a Recurrent High-Order Neural Network (RHONN). The proposed RHONN serves as an online approximator to compensate for uncertain nonlinear dynamics in a PD-based backstepping controller, enabling the system to handle disturbances, modeling errors, and unmodeled dynamics. Instead of relying on the traditional Extended Kalman Filter (EKF) for RHONN weight adaptation, the neural parameters are updated online using a Super-Twisting Algorithm (STA). As a result, the proposed STA-based learning law provides a simpler and robust covariance-free adaptation mechanism with practical finite-time convergence properties, making it suitable for real-time embedded implementations. The proposed method was evaluated through numerical simulations and implemented on an embedded microcontroller to assess its real-time performance. Simulation results show reductions between 0.04% and 2.04% in steady-state and integral error metrics compared with a tuned PD controller, and improvements up to 25.66% and 23.82% over LQR and MPC in the IMSE index. Experimental results demonstrate good tracking performance, robustness under varying load conditions, and low computational requirements, confirming the practical feasibility.

Abstract In this paper, we propose an attitude takeover control of postcapture combined spacecraft using prescribed performance control (PPC) integrated with time-delay estimation (TDE). In space missions like on-orbit servicing (OOS) and space debris removal (SDR), a service spacecraft and a target form a combined spacecraft in the postcapture phase and it detumbles to achieve a desired orientation. However, model uncertainties such as unknown moment of inertia, center of mass from unknown target, and external disturbances caused by target maneuverability can degrade attitude control performance of the combined spacecraft. To address these challenges, PPC is employed to guarantee control performance in the presence of model uncertainties. In addition, a lumped uncertainty including model uncertainties and external disturbances is estimated using TDE. The proposed method is compared through simulations with conventional model-based PPC, PD control and sliding mode control (SMC). The real-time applicability of the proposed controller was validated through a process-in-the-Loop (PIL) implementation test.

Extensive research has solved the dynamic problems of electromagnetic bearings under 1:1 internal resonance, but the complex nonlinear behavior of a 12-pole variable stiffness system under 1:2 internal resonance has not been explored to a large extent. This paper aims to bridge this gap by providing comprehensive dynamic analysis and PD control strategies for such systems. First, the expression for the electromagnetic force in the system is derived based on electromagnetic theory, applying Newton’s second law and considering the effect of rotor gravity. The resulting differential equations governing the dynamics and control of the magnetic bearing, which include quadratic and cubic terms, are then derived. Next, the relationships between the first-order and second-order natural frequencies are analyzed, taking into account 1:2 internal resonance, primary parameter resonance, and 1/2 subharmonic resonance. A perturbation analysis of the system is performed using the method of multiple time scales, yielding the four-dimensional averaged equations in both polar and Cartesian coordinates, as well as the amplitude–frequency response equations. Finally, numerical simulations reveal distinct vibration modulation patterns: parametric excitation dominates the first-order modal response, whereas external forcing preferentially excites the second-order mode, resulting in anisotropic oscillations along orthogonal axes. Moreover, the differential gain is demonstrated to be a critical factor in the suppression of chaos and bifurcations. These results significantly advance the understanding of the nonlinear dynamic behaviors inherent in active electromagnetic bearing systems.

Multiple studies show conflicting association between APOE polymorphisms and the risk of PDD, yielding inconsistent results. To elucidate, a meta-analysis was conducted using existing articles from Web of Science, PubMed, Cochrane, Google Scholar, Embase, WanFang, and CNKI databases, including case-control studies published up to January 31, 2025. A total of 27 studies (3,115 PD controls and 1,338 PDD cases) were included, with pooled Odds Ratio (ORs) and 95% confidence intervals (CIs) calculated using CMA, Biostat, United States. To assess APOE genotypes and PDD risk, three comparisons were examined: 5 genotypes vs. ε3/3, ε2+/ε4 + vs. ε3/3, and ε4 + vs. ε4−. The ε3/4 (OR = 1.56, 95% CI: 1.25–1.95); ε4 + vs. ε3/3 (OR = 1.52, 95% CI: 1.20–1.93) and ε4 + vs. ε4− (OR = 1.62, 95% CI: 1.39–1.90) genotypes were associated with an increased PDD risk, while ε2 + showed no significant effect (OR = 1.21, 95% CI: 0.88–1.65, p = 0.23). Carriers of ε4 + had a 1.52-fold higher risk compared to ε3/3, and the ε4 + vs. ε4 − comparison revealed a 1.62-fold greater dementia risk in ε4 + carriers. Subgroup analysis by ancestral region confirmed ε4 + as a significant risk factor for PDD across Asian, and Caucasians populations with higher susceptibility in Asian (OR = 1.98, 95% CI: 1.29–3.05) vs. Caucasian (OR = 1.48, 95% CI: 1.11–1.98) populations. Our findings suggest that ε3/4 and ε4/4 increase susceptibility to PDD, underscoring the need for further large-scale studies to validate these associations.

BackgroundParkinson’s disease (PD) is a neurodegenerative disorder characterized by motor and non-motor symptoms. A growing body of evidence shows that a healthy diet, including the Mediterranean diet (MeDi), can slow the disease progression and improve certain motor and non-motor symptoms. There is limited knowledge about the dietary habits of patients with PD in Lithuania. We aimed to investigate nutritional habits, adherence to the MeDi, related demographic and disease-related factors in Lithuanian PD patients.MethodsThe case–control study was conducted at Vilnius University Hospital Santaros Klinikos from 2023 to 2025. A food frequency questionnaire (FFQ) was used to assess dietary habits and construct the MeDi adherence score. Dietary habits and adherence to the MeDi were compared between PD patients and controls. The association between MeDi adherence and the odds of PD, severity of motor and non-motor symptoms, was assessed.ResultsA total of 59 patients with PD and 54 healthy controls (HC) were recruited. The MeDi score in the HC group was slightly higher than in the PD group (31.70 vs. 29.62, p = 0.058), and adherence was low in both groups. Respondents in the PD group consumed fewer potatoes (p = 0.009) and alcohol (p = 0.016) and more fruits (p = 0.012), poultry (p = 0.004), and olive oil (p = 0,024). A statistically significant reduction in PD odds emerged after adjustment for smoking, BMI, and physical activity (OR = 0.92, 95% CI 0.85–0.99, p = 0.042), and remained significant in the fully adjusted model including education and income (OR = 0.90, 95% CI 0.81–0.98, p = 0.025). Higher MeDi adherence was associated with lower odds of pain (OR = 0.86, 95% CI 0.75–0.98), urinary dysfunction (OR = 0.86, 95% CI 0.75–0.99), constipation (OR = 0.88, 95% CI 0.78–0.99), and anxiety (OR = 0.88, 95% CI 0.79–0.99).ConclusionPatients with PD had different dietary habits compared to controls. However, adherence to the MeDi diet was low in the elderly Lithuanian population (> 65 years), including patients with PD. Higher MeDi adherence was associated with lower odds of PD and selected non-motor symptoms in multivariable analyses.

This paper describes the experimental evaluation of a novel identification scheme to determine the parameters of a two degrees of freedom direct-drive robot. The novelty of this scheme is that it ensures convergence of the estimated parameters to their true parameters, even when the regressor does not satisfy the stringent condition of persistent excitation, using the classical normalised gradient descendent algorithm. An indirect adaptive controller using this scheme is also evaluated and compared with a PD plus adaptive compensation controller, where a power consumption index is presented to highlight the superior performance of the proposed scheme.

ABSTRACTParkinson's disease (PD) is associated with alterations in both voluntary and reflexive eye movements; however, the characteristics of oculomotor variability across task contexts remain incompletely understood. This study investigated prosaccade, antisaccade and fixation parameters in 15 individuals with early‐ to midstage PD, assessed in the on‐medication state and 15 age‐ and sex‐matched neurologically healthy controls. Eye movements were recorded during structured saccadic tasks and during free‐viewing of dynamic video stimuli using high‐resolution binocular eye tracking. Compared with controls, participants with PD exhibited significantly higher prosaccade error rates and an increased peak velocity‐to‐amplitude ratio. Trial‐to‐trial variability, quantified using coefficients of variation, was consistently elevated in the PD group across multiple saccade parameters. During the video‐viewing condition, changes in saccade metrics following video exposure were observed in the control group but not in the PD group, whereas fixation‐based measures did not reliably differentiate groups. Together, these findings indicate that increased variability and reduced consistency of saccadic execution are prominent features of oculomotor control in PD without freezing of gait, particularly during reflexive saccade tasks. The results underscore the value of variability‐based analyses for probing sensorimotor control in PD and motivate future work to examine their task dependence, longitudinal stability and relevance across disease stages.

Adaptive motion control for autonomous mobile robots: A comparative study of robust tracking under dynamic uncertainties

• Aeroelastic model is used to model the flexible wing and reinforcement learning is used to train a policy for turbulent gust rejection. • Three sets of observations and two actuator time constants are investigated. • Inclusion of current control surface deflection to the observation does not result in any significant performance improvement. • Observing the incoming gust velocity through an alpha probe significantly improves the performance of the controller. • Actuator with lower time constant is more responsive to the commands, resulting in better control performance. • Reinforcement learning policy out-performs PD and LQG controllers. In this paper, reinforcement learning is implemented to train a neural network-based feedback controller for turbulent gust rejection on a flexible wing. The aeroelastic model of the wing is modeled by coupling the unsteady vortex lattice method for unsteady aerodynamics with finite-element based structural dynamics. Thereafter, reinforcement learning via the proximal policy optimization algorithm is used to train a neural network to minimize gust-induced tip deflections by directly manipulating the control surface. Results from simulation show that the trained policies are able to reduce the mean-squared-error in the tip displacements compared to the open-loop responses. Additionally, observing the incoming gust magnitudes, measured by an alpha probe, significantly improves the performance of the controller by allowing it to take preemptive actions. The reinforcement learning policy is able to better adapt to and learn complex dynamics, resulting in better overall performance when compared to PD and LQG controllers.

Optimized virtual synchronous generator control via three-term model predictive control for enhanced disturbance adaptability

Jumping is a critical capability for quadruped robots, especially for navigating obstacles and gaps in complex environments. For successful jump, accurate trajectory tracking and robust feedback mechanism are essential, as cumulative deviations from the desired jumping trajectory can lead to instability or landing failure. Existing controllers often rely on fixed joint-level PD control or simplified inverse dynamics, which often fall short in tracking accuracy and robustness. In this paper, we propose a phase-aware iterative Linear Quadratic Regulator (iLQR) framework tailored for dynamic quadruped jumping tasks. By segmenting the jumping motion into distinct phases, we define phase-wise optimal control problem that respects the unique characteristics and requirements of each stage. Moreover, by leveraging a planar full-body dynamics of quadruped in each iLQR sub-problem, we derive a tracking controller consisting time-varying, full-state feedback gains, which shows better performance in tracking accuracy and disturbances rejection over traditional baseline controllers. Extensive simulation and hardware experiments on the Deeprobotics Lite3 quadruped validate the effectiveness and reliability of our proposed method in a number of dynamic jumping scenarios.

BackgroundAlpha-synuclein can be detected in skin biopsies of individuals with synucleinopathies. However, quantitative data of total and phosphorylated Serine 129 (pS129) alpha-synuclein in skin biopsies are scarce.ObjectiveWe aimed to investigate the biomarker potential of quantitative total and pS129 alpha-synuclein measurements in skin biopsies from people with synucleinopathies and controls.MethodsWe developed and validated AlphaLISA™ immunoassays to determine total and pS129 alpha-synuclein concentrations. Postmortem skin biopsies of Parkinson's disease (PD: n = 18), Dementia with Lewy bodies (DLB: n = 3), Multiple System Atrophy (MSA: n = 5) and control (n = 5) subjects were collected at the cervical vertebra C7. Brain tissues (middle temporal gyrus and substantia nigra) were collected from these same cases. In addition, skin biopsies of controls (n = 20) and PD cases (n = 40) were obtained from the ProPark cohort.ResultsTotal and pSer129 alpha-synuclein could be robustly detected and quantified in all skin samples. We observed a trend towards increased total (+58%, p = 0.055) and pS129 (+131%, p = 0.060) alpha-synuclein skin concentrations in synucleinopathy cases compared to controls. We found no correlations between pS129 alpha-synuclein concentrations in paired brain and skin tissues from the same donors. pS129 alpha-synuclein concentrations were similar for clinical PD cases and controls and there was no correlation with motor symptom severity (UPDRS-III).ConclusionsThese findings highlight that total and pS129 alpha-synuclein can be biochemically quantified in skin biopsies, but warrant further validation and investigation to asses its potential as a diagnostic biomarker in clinical cohorts.

DC–DC buck converters are inherently nonlinear systems that often operate under dynamically changing conditions, parameter uncertainties, and external disturbances, posing significant challenges for conventional control strategies. This paper introduces a novel cascaded proportional–integral and proportional–derivative (PI–PD) controller architecture, in which all controller parameters are optimally tuned using the recently developed Electric Eel Foraging Optimizer (EEFO), a bio-inspired metaheuristic algorithm modeled on the electrolocation and hunting behaviors of electric eels. The proposed control structure uniquely integrates a dual-loop configuration: the inner PI loop eliminates steady-state error, while the outer PD loop enhances dynamic response and mitigates rapid transient fluctuations. This cascaded arrangement enables decoupled tuning of steady-state and transient characteristics, offering superior control flexibility compared to conventional single-loop PID designs. To calibrate the controller, EEFO is employed to minimize a composite performance objective function that simultaneously considers settling time and overshoot, ensuring well-damped and rapid system behavior. A comprehensive set of simulation experiments was conducted in a MATLAB/Simulink environment to evaluate the proposed method against multiple benchmark algorithms—including the flood algorithm, gazelle optimization algorithm, and artificial hummingbird algorithm—as well as classical PID, PID acceleration (PIDA), and fractional-order PID (FOPID) controllers optimized by state-of-the-art metaheuristics. Across all key performance metrics—including rise time, settling time, percentage overshoot, peak time, and steady-state error—the EEFO-tuned cascaded PI–PD controller demonstrated consistently superior results, achieving near-zero overshoot, ultra-fast convergence, and minimal output deviation. Beyond nominal conditions, extensive robustness analyses were conducted to validate the controller’s effectiveness under realistic disturbances, such as abrupt load changes, high-frequency measurement noise, time-delay effects in feedback channels, and ±10%–15% parametric variations in inductance and capacitance. In all scenarios, the controller retained stable output regulation, confirming its resilience and practical viability. To the best of our knowledge, this is the first study to deploy a cascaded PI–PD control structure specifically designed for DC–DC buck converters and optimized using the EEFO algorithm. The integration of a biologically inspired optimization framework with a decoupled dual-loop control scheme offers both architectural and algorithmic novelty. The proposed strategy addresses critical demands in nonlinear converter regulation and provides a robust, high-performance solution suitable for dynamic and uncertain power electronic environments.

ABSTRACT Recent research in control theory has underscored the growing interest in fractional order plants that include dead time. However, tuning fractional order controllers remains challenging due to the numerous adjustable parameters and the absence of systematic tuning approaches. Managing integrating industrial processes with dead time is especially difficult. Control structures with dual-loop configuration are more effective than traditional proportional-integral-derivative (PID) controllers in unity feedback loops for these types of processes. In this paper, a dual-loop fractional order PID − PD control structure combined with Smith Predictor (SP) is presented for fractional order processes. The method is derived from H 2 control theory in the frequency domain. It begins by exploring the constraints associated with internal stability and the asymptotic behavior of the closed-loop system. Following this, a novel procedure for analytically designing the controller is established, resulting in straightforward design formulas. To mitigate the instability that may arise from SP control structures, particularly as a result of fluctuations in time delay, the design focuses on ensuring the robustness of the closed-loop system against uncertainties. The suggested approach relies on performance specifications in the frequency domain and adheres to the maximum sensitivity ( M s ) criterion.