Time-delay Stability Research Articles

Detecting causalities between strongly coupled dynamical systems

A new version of the convergent cross mapping is proposed to detect causalities from records for strongly coupled nonlinear dynamical systems, where the mutual entropy is used to measure nonlinear correlations, and the time delay stability is adopted to filter out false identifications. Calculations on various deterministic dynamic systems show that it is applicable not only to strongly coupled systems but also to non-interacting systems influenced by a common environment. Compared with the original version of convergent cross mapping, under strong couplings our proposed method has significantly higher accuracy, and is more robust to coupling strength. As a typical example, it is used to detect the causal effects between arterial blood pressure (ABP) and intracranial pressure (ICP) of patients diagnosed with traumatic brain injury (TBI). A mono-directional causality from ICP to ABP is identified.

Time-delayed control of a nonlinear self-excited structure driven by simultaneous primary and 1:1 internal resonance: analytical and numerical investigation

<p>Main objective of this research to eliminate the resonant vibrations and stabilize the unstable motion of a self-excited structure through the implementation of an innovative active control strategy. The control strategy coupling the self-excited structure with a second-order filter, which feedback gain $ \lambda $ and control gain $ \beta $, as well as a first-order filter, which feedback gain $ \delta $ and control gain $ \gamma $. The coupling of the second-order filter to establish an energy bridge between the structure and the filter to pump out the structure's vibration energy to the filter. In contrast, the primary purpose of coupling the first-order filter to stabilize the closed loop by adjusting the damping of the system using the control keys $ \delta $ and $ \gamma $. Accordingly, the mathematical model of the proposed control system formulated, incorporating the closed-loop time delay $ \tau $. An analytical solution for the system model obtained, and a nonlinear algebraic system for the steady-state dynamics of the controlled structure extracted. The system's bifurcation characteristics analyzed in the form of stability charts and response curves. Additionally, the system's full response simulated numerically. Findings the high performance of the introduced controller in eliminating the structure's resonant vibrations and stabilizing non-resonant unstable motion. In addition, analytical and numerical investigations revealed that the frequency band within which the second-order filter can absorb the structure's resonant oscillation relies on the algebraic product of $ \beta $ and $ \lambda $. Furthermore, it was found that the equivalent damping of the system depends on the algebraic product of $ \gamma $ and $ \delta $, which can be employed to stabilize the negatively damped self-excited systems. Finally, it reported that although the loop delay can potentially degrade vibration control performance, the time-delay stability margin is nonlinearly proportional to the product of $ \gamma $ and $ \delta $. This finding that increasing the value of $ \gamma \times \delta $ can compensate for the adverse effects of loop delay on both system stability and vibration suppression efficiency.</p>

Physiologic Network-Based Brain-Heart Interaction Quantification During Visual Emotional Elicitation.

In recent years, there has been a surge in interest regarding the intricate physiological interplay between the brain and the heart, particularly during emotional processing. This has led to the development of various signal processing techniques aimed at investigating Brain-Heart Interactions (BHI), reflecting a growing appreciation for their bidirectional communication and influence on each other. Our study contributes to this burgeoning field by adopting a network physiology approach, employing time-delay stability as a quantifiable metric to discern and measure the coupling strength between the brain and the heart, specifically during visual emotional elicitation. We extract and transform features from EEG and ECG signals into a 1 Hz format, facilitating the calculation of BHI coupling strength through stability analysis on their maximal cross-correlation. Notably, our investigation sheds light on the critical role played by low-frequency components in EEG, particularly in the δ , θ , and α bands, as essential mediators of information transmission during the complex processing of emotion-related stimuli by the brain. Furthermore, our analysis highlights the pivotal involvement of frontal pole regions, emphasizing the significance of δ - θ coupling in mediating emotional responses. Additionally, we observe significant arousal-dependent changes in the θ frequency band across different emotional states, particularly evident in the prefrontal cortex. By offering novel insights into the synchronized dynamics of cortical and heartbeat activities during emotional elicitation, our research enriches the expanding knowledge base in the field of neurophysiology and emotion research.

Stability analysis of low-voltage direct current system with time-varying delay

ABSTRACT This paper investigates the stability analysis of LVDC systems with time-varying delay. A hierarchical control model is constructed for LVDC systems subjected to time-varying delay. Furthermore, a novel Lyapunov-Krasovskii functional is developed by introducing the augmented terms and the delay-dependent matrices. To reduce the conservativeness of stability criteria, the constructed functional is effectively estimated by using the quadratic function inequalities and composite relaxation integral inequalities. Finally, the simulation results are given to validate the proposed method with a higher time-delay stability margin of LVDC systems.

Dynamic networks of cortico-muscular interactions in sleep and neurodegenerative disorders.

The brain plays central role in regulating physiological systems, including the skeleto-muscular and locomotor system. Studies of cortico-muscular coordination have primarily focused on associations between movement tasks and dynamics of specific brain waves. However, the brain-muscle functional networks of synchronous coordination among brain waves and muscle activity rhythms that underlie locomotor control remain unknown. Here we address the following fundamental questions: what are the structure and dynamics of cortico-muscular networks; whether specific brain waves are main network mediators in locomotor control; how the hierarchical network organization relates to distinct physiological states under autonomic regulation such as wake, sleep, sleep stages; and how network dynamics are altered with neurodegenerative disorders. We study the interactions between all physiologically relevant brain waves across cortical locations with distinct rhythms in leg and chin muscle activity in healthy and Parkinson's disease (PD) subjects. Utilizing Network Physiology framework and time delay stability approach, we find that 1) each physiological state is characterized by a unique network of cortico-muscular interactions with specific hierarchical organization and profile of links strength; 2) particular brain waves play role as main mediators in cortico-muscular interactions during each state; 3) PD leads to muscle-specific breakdown of cortico-muscular networks, altering the sleep-stage stratification pattern in network connectivity and links strength. In healthy subjects cortico-muscular networks exhibit a pronounced stratification with stronger links during wake and light sleep, and weaker links during REM and deep sleep. In contrast, network interactions reorganize in PD with decline in connectivity and links strength during wake and non-REM sleep, and increase during REM, leading to markedly different stratification with gradual decline in network links strength from wake to REM, light and deep sleep. Further, we find that wake and sleep stages are characterized by specific links strength profiles, which are altered with PD, indicating disruption in the synchronous activity and network communication among brain waves and muscle rhythms. Our findings demonstrate the presence of previously unrecognized functional networks and basic principles of brain control of locomotion, with potential clinical implications for novel network-based biomarkers for early detection of Parkinson's and neurodegenerative disorders, movement, and sleep disorders.

EEG-based Emotion Recognition Using Sub-Band Time-Delay Correlations.

Emotion recognition is a challenging task with many potential applications in psychology, psychiatry, and human-computer interaction (HCI). The use of time-delay information in the controlled time-delay stability (cTDS) algorithm can help to capture the temporal dynamics of EEG signals, including sub-band information and bi-directional coupling that can aid in emotion recognition and identification of specific connectivity patterns between brain rhythms. Incorporating EEG frequency bands can be used to design better emotion recognition systems. This paper evaluates the cTDS algorithm for binary classification tasks of arousal and valence using EEG sub-band signals. This method achieved a high accuracy of 91.1% for arousal and 91.7% for valence based on one electrode recording site at Fp1. The cTDS algorithm is a promising approach to analyzing brain network interactions. It can be particularly applicable to arousal and valence classification tasks, especially within a complex, multimodal feature space associated with understanding psychiatric disorders and HCI applications.

Graph-based learning of nonlinear physiological interactions for classification of emotions

Emotion recognition has been drawing the attention of researchers and practitioners in recent years. While various research studies show successful applications to recognize and distinguish emotions based on physiological responses using machine learning techniques, less research to date is focused on the network properties of physiological interactions under different emotional states. To this end, we propose a multi-modal graph learning framework to quantify the interactions among physiological systems and present representative networks associated with emotional states. More specifically, we introduce a novel information-theoretic-based time delay stability to quantify complex interactions between physiological modalities. We test our quantification approach on three publicly available benchmark databases for emotion recognition and demonstrate the comparative performances of measuring the interactions of physiological systems in response to emotional states. Finally, we present the visualization of multi-modal physiological network topology, which may be useful for emotional interpretations in practice.

Bifurcation analysis of a two–dimensional p53 gene regulatory network without and with time delay

<abstract><p>In this paper, the stability and bifurcation of a two–dimensional p53 gene regulatory network without and with time delay are taken into account by rigorous theoretical analyses and numerical simulations. In the absence of time delay, the existence and local stability of the positive equilibrium are considered through the Descartes' rule of signs, the determinant and trace of the Jacobian matrix, respectively. Then, the conditions for the occurrence of codimension–1 saddle–node and Hopf bifurcation are obtained with the help of Sotomayor's theorem and the Hopf bifurcation theorem, respectively, and the stability of the limit cycle induced by hopf bifurcation is analyzed through the calculation of the first Lyapunov number. Furthermore, codimension-2 Bogdanov–Takens bifurcation is investigated by calculating a universal unfolding near the cusp. In the presence of time delay, we prove that time delay can destabilize a stable equilibrium. All theoretical analyses are supported by numerical simulations. These results will expand our understanding of the complex dynamics of p53 and provide several potential biological applications.</p></abstract>

Prescribed-time cooperative guidance with time delay

AbstractIn view of the cooperative guidance problem with time delay, this paper proposes a two-stage time-delay prescribed-time cooperative guidance law in the three-dimensional (3D) space. In the first stage, by introducing a time scaling function and time-delay consensus, the proposed cooperative guidance law can overcome the negative influence of time delay to guaranteed the desired convergence performance. Derived from the Lyapunov convergence analysis, the time-delay stability of the first stage can be ensured and the convergence time can be described as the relationship between delayed time and mission-assigned convergence time. Then, taking the prescribed-time-related convergence time as the switching point, the second stage begins with suitable initial conditions and all interceptors are governed by proportional navigation guidance. Finally, comparative simulations are performed to demonstrate the effectiveness and superiority of the proposed time-delay guidance law.

Time-delay Stability Analysis of the Distributed Economic Dispatch System of an Active Distribution Network

Distributed economic dispatch system has gained a lot of attention because it has better robustness, faster computation and less communication burden, but the system stability in the presence of communication delay is still researched at the fundamental level. By introducing the concept of virtual distributed generation, this paper develops a fully version of the distributed economic dispatching strategy for an active distribution network, and establishes the stability analysis model of the distribution network considering symmetric and asymmetric communication time-delay. The Lyapunov stability theorem and the free weight matrix method are comprehensively applied to derive the time-delay stability criterion of the system and solve the delay stability margin with a lower conservativeness. The distributed economic dispatch strategy in this paper under various operating conditions, the delay stability analysis method and delay stability margin solution are verified by the simulation results on an active distribution network modified based on the IEEE-14 system.

Dynamic networks of physiologic interactions of brain waves and rhythms in muscle activity.

The brain plays a central role in facilitating vital body functions and in regulating physiological and organ systems, including the skeleto-muscular and locomotor system. While neural control is essential to synchronize and coordinate activation of various muscle groups and muscle fibers within muscle groups in relation to body movements and distinct physiologic states, the dynamic networks of brain-muscle interactions have not been explored and the complex regulatory mechanism of brain-muscle control remains unknown. Here we present a first study of network interactions between brain waves at different cortical locations and peripheral muscle activity across key physiologic states - wake, sleep and distinct sleep stages. Utilizing a novel approach based on the Network Physiology framework and the concept of time delay stability, we find that for each physiologic state the network of cortico-muscular interactions is characterized by a specific hierarchical organization of network topology and network links strength, where particular brain waves are main mediators of interaction and control of muscular activity. Further, we uncover that with transition from one physiological state to another, the brain-muscle interaction network undergoes marked reorganization in the profile of network links strength, indicating a direct association between network structure and physiological state and function. The pronounced stratification in brain-muscle network characteristics across sleep stages is consistent for chin and leg muscle groups and persists across subjects, indicating a remarkable universality and a previously unrecognized basic physiologic mechanism that regulates muscle activity even during rest and in the absence targeted direct movement. Our findings demonstrate previously unrecognized coordination between brain waves and activation of different muscle fiber types within muscle groups, laws of brain-muscle cross-communication and principles of network integration and control. These investigations demonstrate the potential of network-based biomarkers for classification of distinct physiological states and conditions, for the diagnosis and prognosis of neurodegenerative, movement and sleep disorders, and for developing efficient treatment strategies.

Cascade Modular Path Following Control Strategy for Gantry Virtual Track Train: Time-Delay Stability and Forward Predictive Model

The virtual track train (VTT) is a new type of urban transit system with all-wheel active steering technology utilized to make the long combination vehicle run along the planned route (virtual track) like railway vehicles along physical track. This paper proposed a scalable cascade modular path following control strategy for the newly designed gantry virtual track train (G-VTT) based on preview & tracking controller, focusing on the lateral control for low-speed turning manoeuvre. The mathematical characteristics of the 2-norm was utilized to establish a local tracking objective function (LTOF) to find the optimal lateral acceleration (OLA) of the tracking point in the preview window and calculate the optimal wheel steering angle (WSA) to achieve the optimal local trajectory. Considering the lag time <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> , the yaw stability and path following ability of the vehicle are contradictory in some cases, so the damping adjustment factor <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</i> was proposed to optimize the damping characteristics and the time-delay stability factor <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ξ</i> was proposed to evaluate the time-delay stability. The influence of the key parameters, preview time <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ξ</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</i> , on the stability and following ability were analyzed by frequency domain analysis method, and the optimal region of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ξ</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</i> was obtained to instruct the configuration of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> . Finally, steady-state and dynamic forward predictive models were proposed to compensate for the oscillation caused by the lag time. The effectiveness of the proposed control strategy, forward predictive models and the configuration method of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> were verified by theoretical analysis and numerical simulation.

Distributed Dynamic Event-Triggered Power Management Strategy for Global Economic Operation in High-Power Hybrid AC/DC Microgrids

With the increasing integration of distributed generators (DGs), renewable energy sources (RESs), and energy storage systems (ESSs) in the hybrid AC/DC microgrid (HMG), the paralleled bidirectional interlinking converters (BILCs) play a significant role in the economic power interaction (EPI) between AC and DC subgrids. This paper proposes a distributed dynamic event-triggered control (ETC) strategy for the HMG to realize the global economic operation. In each subgrid, economic power sharing (EPS) among DGs can be achieved by economic droop controllers with no global information. The proposed distributed dynamic ETC includes a continuous-time part and a discrete-time part, where the former part realizes the EPI between two subgrids, while the latter part ensures the proportional power sharing between BILCs. Thus, a particular over-stressed BILC can be avoided, and system reliability and scalability are significantly improved. Additionally, only data from neighboring BILCs at the event-triggered times are involved, which greatly minimizes the communication burden and costamong BILCs. Furthermore, the asymptotic stability of the closed-loop system dynamics and the communication time delay stability are proven. Finally, the hardware-in-loop experimental results are conducted to demonstrate the effectiveness of the proposed control strategy.

Changes in physiological network connectivity of body system in narcolepsy during REM sleep.

Narcolepsy is marked by pathologic symptoms including excessive daytime drowsiness and lethargy, even with sufficient nocturnal sleep. There are two types of narcolepsy: type 1 (with cataplexy) and type 2 (without cataplexy). Unlike type 1, for which hypocretin is a biomarker, type 2 narcolepsy has no adequate biomarker to identify the causality of narcoleptic phenomenon. Therefore, we aimed to establish new biomarkers for narcolepsy using the body's systemic networks. Thirty participants (15 with type 2 narcolepsy, 15 healthy controls) were included. We used the time delay stability (TDS) method to examine temporal information and determine relationships among multiple signals. We quantified and analyzed the network connectivity of nine biosignals (brainwaves, cardiac and respiratory information, muscle and eye movements) during nocturnal sleep. In particular, we focused on the differences in network connectivity between groups according to sleep stages and investigated whether the differences could be potential biomarkers to classify both groups by using a support vector machine. In rapid eye movement sleep, the narcolepsy group displayed more connections than the control group (narcolepsy connections: 24.47±2.87, control connections: 21.34±3.49; p=0.022). The differences were observed in movement and cardiac activity. The performance of the classifier based on connectivity differences was a 0.93 for sensitivity, specificity and accuracy, respectively. Network connectivity with the TDS method may be used as a biomarker to identify differences in the systemic networks of patients with narcolepsy type 2 and healthy controls.

Stability Analysis of Grid-connected Inverter System Containing Virtual Synchronous Generator under Time Delay and Parameter Uncertainty

Virtual synchronous generator (VSG) control is an effective way to increase the equivalent inertia of grid connected inverter system and improve the stability of the power grid. However, the influence of this control on the stability of the whole system with time delay and parameter uncertainty should be researched further. In this paper, the state space model of the whole grid connected inverter system adopting VSG under control time-delay and parameter uncertainty is established. The Lyapunov functional with time-delay is constructed by employing time-delay dependent free weight matrix transformation. Furthermore, Lyapunov stability theorem is used to obtain the more conservative time-delay stability conditions with and without parameter uncertainty, linear matrix inequality is used to calculate the time-delay stability margin of the system under uncertainty, and the influence of the parameters of VSG control on the delay stability margin is analyzed. The validity of the proposed stability analysis and stability margin solution method are verified by simulation examples.

An Effective Joint Soft-Sensing Strategy for Multi-Information under Diverse Vehicle Driving Scenarios

A variety of accurate information inputs are of great importance for automotive control. In this paper, a novel joint soft-sensing strategy is proposed to obtain multi-information under diverse vehicle driving scenarios. This strategy is realized by an information interaction including three modules: vehicle state estimation, road slope observer and vehicle mass determination. In the first module, a variational Bayesian-based adaptive cubature Kalman filter is employed to estimate the vehicle states with the time-variant noise interference. Under the assumption of road continuity, a slope prediction model is proposed to reduce the time delay of the road slope observation. Meanwhile, a fast response nonlinear cubic observer is introduced to design the road slope module. On the basis of the vehicle states and road slope information, the vehicle mass is determined by a forgetting-factor recursive least square algorithm. In the experiments, a contrasted strategy is introduced to analyse and evaluate performance. Results declare that the proposed strategy is effective and has the advantages of low time delay, high accuracy and good stability.

Network Physiology of Cortico-Muscular Interactions.

Skeletal muscle activity is continuously modulated across physiologic states to provide coordination, flexibility and responsiveness to body tasks and external inputs. Despite the central role the muscular system plays in facilitating vital body functions, the network of brain-muscle interactions required to control hundreds of muscles and synchronize their activation in relation to distinct physiologic states has not been investigated. Recent approaches have focused on general associations between individual brain rhythms and muscle activation during movement tasks. However, the specific forms of coupling, the functional network of cortico-muscular coordination, and how network structure and dynamics are modulated by autonomic regulation across physiologic states remains unknown. To identify and quantify the cortico-muscular interaction network and uncover basic features of neuro-autonomic control of muscle function, we investigate the coupling between synchronous bursts in cortical rhythms and peripheral muscle activation during sleep and wake. Utilizing the concept of time delay stability and a novel network physiology approach, we find that the brain-muscle network exhibits complex dynamic patterns of communication involving multiple brain rhythms across cortical locations and different electromyographic frequency bands. Moreover, our results show that during each physiologic state the cortico-muscular network is characterized by a specific profile of network links strength, where particular brain rhythms play role of main mediators of interaction and control. Further, we discover a hierarchical reorganization in network structure across physiologic states, with high connectivity and network link strength during wake, intermediate during REM and light sleep, and low during deep sleep, a sleep-stage stratification that demonstrates a unique association between physiologic states and cortico-muscular network structure. The reported empirical observations are consistent across individual subjects, indicating universal behavior in network structure and dynamics, and high sensitivity of cortico-muscular control to changes in autonomic regulation, even at low levels of physical activity and muscle tone during sleep. Our findings demonstrate previously unrecognized basic principles of brain-muscle network communication and control, and provide new perspectives on the regulatory mechanisms of brain dynamics and locomotor activation, with potential clinical implications for neurodegenerative, movement and sleep disorders, and for developing efficient treatment strategies.

Interface design and dielectric response behavior of SiO2/PB composites with low dielectric constant and ultra-low dielectric loss

The resistance-capacitance time delay, crosstalk interference and poor dimensional stability would damage the signal intact transmission. However, it was still a tough challenge to fabricate an interlayer dielectric material with low dielectric constant and low dielectric loss as well as excellent dimensional stability at the same time. In this work, for the first time, we constructed novel silicon dioxide/polybutadiene (SiO2/PB) composites with different SiO2 morphologies and surface function groups, which showed extremely low dielectric constant and dielectric loss in both low and high frequency bands. The crosslinking reaction between surface surficial vinyl groups of the SiO2 and PB macromolecules tremendously suppressed the interfacial polarization of the SiO2/PB composites. Therefore, the SiO2/PB composites exhibited low dielectric constant of 2.66 and ultralow dielectric loss of 0.0022 at high frequency band (10 GHz), and the coefficient of thermal expansion decreased to 154 × 10−6 /°C, which was three orders of magnitude less than that of the pure PB. The interface polarization response behavior of SiO2/PB composites was also elucidated by double layer theory, and the "resonance" phenomenon was firstly proposed to enhance the interface polarization. This work could provide a robust and simple strategy for the design of composites with ultralow dielectric loss and excellent dimensional stability for high-speed communication in different frequency bands.

Time delay stability analysis for vibration suppression of a smart cantilever beam with hysteresis property

A stability analysis for a smart beam with an adaptive controller is presented in the paper when considering the time delay phenomenon. With the Lagrange equation and the assumed modes method, a dynamical model is constructed to describe the hysteresis nonlinearity of the smart beam. By simulation and experiment, the nonlinear model is proved effectively using the strain response near the root of the beam when the piezoelectric actuator is applied on with a chirp voltage. Based on the dynamical model, a stability analysis method is proposed with an eigenmatrix in the discrete control system. Through some simulation verifications, it is concluded that a proper time delay will be useful to improve the stability of the smart beam with an adaptive controller. Furthermore, it is verified by the experiments that the free vibration amplitude of the smart beam with an artificial time delay 0.05 s is smaller compared with that at no time delay.

Reach on damping control and stability analysis of vehicle with double time-delay and five degrees of freedom

Suspension system is one of the important parts of a vehicle, which is used to buffer the impact of uneven road to the body and passengers, so the suspension system has an important impact on the safety and ride comfort of the vehicle. In order to improve the safety and comfort of passengers and vehicles, in this paper a five-degree-of-freedom half car model is established, and the uncertainty of the model and the time-delay of the control are considered. The dynamic response of vehicle body acceleration root mean square, passenger acceleration root mean square, displacement root mean square and vehicle body pitch acceleration root mean square are selected as optimization objectives. The time-delay control parameters are determined by chaos particle swarm optimization algorithm. The time-delay stability of the suspension control system is analyzed by frequency-domain scanning method to ensure the stability of the time-delay control system. Finally, by establishing the simulation model of the active suspension system with double time-delay feedback control, the response characteristics of the suspension system with double time-delay active feedback control to simple harmonic excitation and random excitation input are analyzed. The results show that under the premise of ensuring the system stability, the active suspension system with double time-delay feedback control has good and obvious controlling and damping effect on the body and seats.