Root Trajectory Research Articles

Dynamic Analysis and Equivalent Modeling for a Four-Axle Vehicle

This paper focuses on a comprehensive study of a four-axle vehicle, including dynamics analysis, equivalent modeling methods, and their comparison. Firstly, a linear two-degree lateral dynamic model is established, which has four drive axles and two steer axles. Secondly, the mathematical transfer function expressions for the yaw rate and the centroid sideslip angle were derived on the basis of the model. The steady-state parameters, such as yaw rate gain Gγss, centroid sideslip angle gain Gβss, stability factor Kn, equivalent axial distance ln, and equivalent centroid sideslip angle coefficient Kn’ were obtained by using the transfer functions. Then, the steady-state and transient characteristics are roundly discussed, including steady-state parameters, system root trajectory, frequency domain, and time domain. Some recommendations for the four-axle vehicle’s parameter design are also given. Finally, for a more simple and efficient analysis of response characteristics of four-axle vehicles and even n (n > 4) axle vehicles, the equivalent model is developed for the four-axle vehicle, and comprehensive analyses are presented with four equalization methods, which are based on the inner heart of the approximation triangle, the outer heart of the approximation triangle, the center of gravity of the approximation triangle and the compensation point. Following a thorough analysis of the four, it is determined that the inner approximation triangle solution approach is most suited for four-axle vehicles.

Asymmetric instability of flow-induced vibration for elastically mounted cube at moderate Reynolds numbers

The asymmetric instability in two streamwise orthogonal planes for three-dimensional flow-induced vibration (FIV) of an elastically mounted cube at a moderate Reynolds number of 300 is numerically investigated in this paper. The full-order computational fluid dynamics method, data-driven stability analysis via the eigensystem realization algorithm and the selective frequency damping method and total dynamic mode decomposition (TDMD) are applied here to explore this problem. Due to the unsteady non-axisymmetric wakefield formed for flow passing a stationary cube, the FIV response was found to exhibit separate structural stability and oscillations (including lock-in and galloping behaviour) in the two different streamwise orthogonal planes while the body is released. The initial kinetic energy accompanying the release of the cube could destabilize the above-mentioned structural stability. The observed FIV asymmetric instability is verified by the root trajectory of the structural mode obtained via data-driven stability analysis. The stability of the structural modes dominates regardless of whether the structural response oscillates significantly in various (reduced) velocity ranges. Further TDMD analysis on the wake structure, accompanied by the time–frequency spectrum of time-history structural displacements, suggested that the present FIV unit with galloping behaviour is dominated by the combination of the shifted base-flow mode, structure modes and several harmonics of the wake mode.

Pitching Stabilization Control for Super Large Ships Based on Double Nonlinear Positive Feedback under Rough Sea Conditions

Due to the rapid development of a global navigation satellite system and the rapid growth of ships, the traditional control algorithms are not suitable; hence, the longitudinal rocking phenomenon generated by external disturbances is more serious when a ship is sailing. This paper takes a mathematical model of the super large oil tanker “KVLCC2”’s longitudinal motion as the controlled plant, establishing a multi-input multi-output instability control system, using the root trajectory shaping method and a weighting matrix to ensure the stability of its transfer function’s mathematical model. An improved closed-loop gain-shaping algorithm is utilized to design a simple robust controller. And a dual nonlinear positive feedback control algorithm is added to the control system to further improve the controller’s pitching stabilization performance and reduce the controller’s output energy. In order to verify that the controller has a consistently strong robustness, simulation experiments are carried out by adding a level 6, 7 and 8 wind wave model and a perturbation link to the control system, respectively. The results show that when the value of the hysteresis constant is taken as 0.25, the output values of the heave displacement and the pitch angle are greatly reduced, and the longitudinal rocking phenomenon is significantly improved. The dual nonlinear positive feedback control algorithm enhances the ship’s pitching stabilization control capability and further reduces the controller’s output energy, which provides technical support for the smooth and efficient sailing of super large ships under changing sea conditions. Combined with a global navigation satellite system, this algorithm provides a new method for pitching stabilization control of super large ships.

Substrate heterogeneities cause root growth trajectories to switch from vertical to oblique.

Hard pans, soil compaction, soil aggregation and stones create physical barriers that can affect the development of a root system. Roots are known to exploit paths of least resistance to avoid such obstacles, but the mechanism through which this is achieved is not well understood. Here, we combined 3D-printed substrates with a high-throughput live imaging platform to study the responses of plant roots to a range of physical barriers. Using image analysis algorithms, we determined the properties of growth trajectories and identified how the presence of rigid circular obstacles affects the ability of a primary root to maintain its vertical trajectory. Results showed the types of growth responses were limited, both vertical and oblique trajectories were found to be stable and influenced by the size of the obstacles. When obstacles were of intermediate sizes, trajectories were unstable and changed in nature through time. We formalised the conditions for root trajectory to change from vertical to oblique, linking the angle at which the root detaches from the obstacle to the root curvature due to gravitropism. Exploitation of paths of least resistance by a root may therefore be constrained by the ability of the root to curve and respond to gravitropic signals.

Dynamic impact analysis of the time-delay levitation control system on maglev vehicle system after adding smith predictor

The time delay (TD) in the levitation control system significantly affects the dynamic performance of the closed-loop system in electromagnetic suspension (EMS) maglev vehicles. Excessive TD can cause levitation instability, making it essential to explore effective mitigation methods. To address this issue, a Smith Predictor (SP) is integrated into the traditional PID levitation control system. The combination of theoretical analysis and numerical simulation is employed to assess the stability of the time-delay levitation control system after the integration of the Smith Predictor. Theoretical analysis reveals that when TD exceeds a critical threshold, the levitation system becomes unstable. The addition of SP alters the root trajectory of the system characteristic equation from positive to negative, and recovers the levitation system to stable status. Assuming complete knowledge of the dynamic system, the TD compensation value in the SP becomes a key parameter that determines its performance. A minimum effective value (MEV) for TD compensation is identified, correlating with the system's stability region. Under the influence of TD, more complex systems and higher running speeds of the maglev vehicle lead to a narrower stable region and a larger MEV for TD compensation. Given the simulation parameters in this paper, with a system TD of 15 ms and a maximum vehicle speed of 160 km/h, the MEV for TD compensation in the SP should be set at 12 ms.

Traffic flow bifurcation control of autonomous vehicles through a hybrid control strategy combining multi-step prediction and memory mechanism with PID

This paper is committed to capturing the dynamic behaviors of homogenous flow of autonomous vehicles (AVs), and exploring the control strategies to improve traffic conditions, which can alleviate traffic congestion and improve traffic efficiency. Firstly, a car-following model of AVs considering real-time driving state is established. Secondly, based on bifurcation theory and stability theory, bifurcation analysis is carried out and the relationship between bifurcation and stability is revealed. In order to suppress the bifurcation and improve the stability, a controller considering multi-step prediction and memory mechanism (MPM) is designed, and the root trajectories for eigenvalues and stable time length of the model controlled by MPM controller are calculated. In response to the limitations of the MPM controller, a hybrid controller including the MPM controller and PID controller is further proposed, and it is found that the model controlled by hybrid controller has greater range of stable bifurcation parameter and stable time length, which means better ability of bifurcation suppression. Finally, the capabilities of the controller proposed in this paper are effectively demonstrated by numerical experiments in MATLAB and simulation experiments in the ROS-Gazebo environment.

Analysis of Grid-Connected Stability of VSG-Controlled PV Plant Integrated with Energy Storage System and Optimization of Control Parameters

In the static stability analysis of the grid-connected photovoltaic (PV) generation and energy storage (ES) system, the grid-side is often simplified using an infinite busbar equivalent, which streamlines the analysis but neglects the dynamic characteristics of the grid, leading to certain inaccuracies in the results. Furthermore, the control parameter design does not consider the coupling relationships among parameters, resulting in arbitrary values and the inability to achieve overall optimality. To address these issues, this paper presents a comprehensive parameter optimization method for the oscillation characteristics of grid-connected PV generation and ES systems in various frequency ranges. Firstly, a detailed modeling of the grid-connected PV generation and ES system is conducted, resulting in the derivation of the system’s small-signal model. This study investigates the impact of parameters related to PV arrays, ES units, and the virtual synchronous generator (VSG) on the system’s characteristic roots using participation factors, sensitivity analysis, and eigenvalue root trajectories. Subsequently, based on the analysis of system root trajectories and the Particle Swarm Optimization (PSO) algorithm, a holistic parameter optimization design method for the system’s oscillation modes is proposed, and the parameter optimization results are obtained. Finally, the accuracy of the theoretical analysis is validated through perturbation testing using both Matlab/Simulink models and the small-signal model.

Flywheel rotor cross-filter tracking feedback control

In light of the matter that the fixed-parameter cross-filter circuit cannot satisfy the phase compensation requirement in the full speed range, a cross-feedback tracking control method based on cross-filter feedback, in which the high-pass cutoff frequency varies with the nutation frequency, is proposed. The stability analysis of the designed control method is carried out by using the phase angle margin and the root trajectory diagram, and the results show that the cross-feedback tracking control can increase the phase angle margin and significantly enhance the stability of the system. Simulation results of the magnetic levitation rotor system show that the designed control method possesses the benefits of high stability, rapid response time, and strong anti-jamming ability.

Improved VSG strategy of grid-forming inverters for supporting inertia and damping

A virtual synchronous generator (VSG) strategy can introduce the rotational inertia and damping characteristics of the synchronous generator to the static inverter, e.g., PV, wind generation, and ESS, which are used to enhance the system frequency support characteristics of the micro-grid. However, under various operations of the VSG, the inertia and damping support capabilities are different, and the conventional VSG strategy with a fixed control coefficient sacrifices a certain degree of dynamic regulation performance with less robustness. This research proposes an improved VSG strategy with adaptive inertia and damping coefficients to increase the flexibility of VSG. To this end, a mathematical model is first established to analyze the impact of various parameters on the characteristics of VSG, and then the root trajectory is used to explore the impacts of various rotational inertia and damping coefficients on the stability of the VSG system. Second, an improved control strategy with adaptive inertia and damping coefficients is proposed based on the characteristics of the second-order system and the system frequency deviations. Finally, a simulation system with the VSG is constructed based on MATLAB/Simulink. The effectiveness of the proposed control strategy is verified by comparing the simulation results to the conventional control strategy with the fixed control coefficients under over-frequency and under-frequency disturbances.

Foot‐constrained spatial‐temporal transformer for keyframe‐based complex motion synthesis

AbstractKeyframe‐based motion synthesis holds significant effects in games and movies. Existing methods for complex motion synthesis often require secondary post‐processing to eliminate foot sliding to yield satisfied motions. In this paper, we analyze the cause of the sliding issue attributed to the mismatch between root trajectory and motion postures. To address the problem, we propose a novel end‐to‐end Spatial‐Temporal transformer network conditioned on foot contact information for high‐quality keyframe‐based motion synthesis. Specifically, our model mainly compromises a spatial‐temporal transformer encoder and two decoders to learn motion sequence features and predict motion postures and foot contact states. A novel constrained embedding, which consists of keyframes and foot contact constraints, is incorporated into the model to facilitate network learning from diversified control knowledge. To generate matched root trajectory with motion postures, we design a differentiable root trajectory reconstruction algorithm to construct root trajectory based on the decoder outputs. Qualitative and quantitative experiments on the public LaFAN1, Dance, and Martial Arts datasets demonstrate the superiority of our method in generating high‐quality complex motions compared with state‐of‐the‐arts.

Active Damping Current Control for Current-Source Inverter-Based PMSM Drives

Current-source inverter (CSI) is considered as a promising candidate in the permanent-magnet synchronous machine (PMSM) drive system for its superior current waveforms and inherent capability of withstanding the short-circuit failure. However, the resonance issue is yielded, resulting from the paralleling of machine inductances and filter capacitors. Very limited control bandwidth can be retained if only considering the passive damping of winding resistance. In this paper, the decoupled active damping current controller in the synchronous frame is investigated. The impedance analysis is performed to make a more intuitive demonstration on the damping mechanism, which reveals that the virtual impedance is not always linearly correlated with the damping gain. The digital delay is crucial for the accurate assessment on the damping effectiveness, and the z-domain discrete model is established to emulate the real circumstance. The quantified analysis is performed on deriving the stable boundaries of controller parameters. The frequency responses and root trajectories are illustrated to validate the conclusions. The optimized parameters are obtained by the proposed tuning procedure, which realizes a balance between the damping effect and the bandwidth performance. Finally, the experiments are performed to verify the proposed control schemes.

Equivalent dimensionality reduction method of doubly‐fed induction generator model and discrimination of Hopf bifurcation type

AbstractThe strong coupling, multivariate, and nonlinear characteristics of the mathematical model of a doubly‐fed induction generator (DFIG) make it challenging to analyze the bifurcation category of the DFIG. Therefore, in this study, we developed a method for analyzing the DFIG model via topologically equivalent dimensionality reduction in the neighborhood of bifurcation values based on the central manifold theorem and realized the discrimination of the Hopf bifurcation type of the DFIG. We established a complex frequency domain model of the DFIG and then calculated the critical open‐loop gain point using the root trajectory method to obtain the relationship between the critical open‐loop gain point and Hopf bifurcation. Thereafter, we performed topological equivalence dimensionality reduction in the neighborhood of the DFIG bifurcation values based on the central manifold theorem. Subsequently, the Hopf bifurcation type of the system was determined using the stability index (first Lyapunov exponent) of the Hopf bifurcation. Finally, a nonlinear time‐domain simulation was used to verify that the four and two‐dimensional DFIGs generate a stable limit cycle by supercritical Hopf bifurcation under the tuning parameter. The results indicated that the proposed method was feasible and effective in the neighborhood of bifurcation values in the state parameter space of DFIG, which provides a theoretical basis for the control of Hopf bifurcation of the DFIG.

Adaptive quasi-proportional resonant control with parameter estimation for PMSM sensorless control

Conventional sliding mode control (SMO) utilizes a sign function and a low-pass filter (LPF) to achieve sensorless control of a permanent magnet synchronous motor (PMSM). However, the introduction of the sign function generates a large number of high harmonics, resulting in a significant system output chattering, whereas the addition of the LPF generates control delays and phase shifts. These degrade the PMSM speed and the position estimation accuracy and reduce the system control performance. To improve the performance of the PMSM, adaptive quasi-proportional resonant sensorless control with parameter estimation (AQPR_PE) is proposed. In this control model, an adaptive quasi-proportional resonance controller is used reduce the system output error and to weaken the chattering phenomenon. Meanwhile, the system PMSM parameters are estimated online and fed back to the control model to improve the system parameter robustness and the control accuracy. The Bode diagram, Popov theory, and root trajectory are used to analyze the stability of AQPR, parameter estimation, and AQPR_PE, respectively. Finally, the effectiveness of the proposed method is demonstrated by experimental validation.

Discriminant and Root Trajectories of Characteristic Equation of Fractional Vibration Equation and Their Effects on Solution Components

The impulsive response of the fractional vibration equation z′′(t)+bDtαz(t)+cz(t)=F(t), b>0,c>0,0≤α≤2, is investigated by using the complex path-integral formula of the inverse Laplace transform. Similar to the integer-order case, the roots of the characteristic equation s2+bsα+c=0 must be considered. It is proved that for any b>0, c>0 and α∈(0,1)∪(1,2), the characteristic equation always has a pair of conjugated simple complex roots with a negative real part on the principal Riemann surface. Particular attention is paid to the problem as to how the couple conjugated complex roots approach the two roots of the integer case α=1, especially to the two different real roots in the case of b2−4c>0. On the upper-half complex plane, the root s(α) is investigated as a function of order α and with parameters b and c, and so are the argument θ(α), modulus r(α), real part λ(α) and imaginary part ω(α) of the root s(α). For the three cases of the discriminant b2−4c: >0, =0 and <0, variations of the argument and modulus of the roots according to α are clarified, and the trajectories of the roots are simulated. For the case of b2−4c<0, the trajectories of the roots are further clarified according to the change rates of the argument, real part and imaginary part of root s(α) at α=1. The solution components, i.e., the residue contribution and the Hankel integral contribution to the impulsive response, are distinguished for the three cases of the discriminant.

Modeling and Stability Analysis of Interconnected System in Weak Grid

Abstract Compared with the conventional grid, the interconnection system has the advantages of high power supply flexibility, high equipment utilization, small installed capacity, and more economical operation. However, unlike in the analysis of stability, the dynamic response of the converter of the interconnected system is faster, and to model its internal structure and control, a new model applicable to it is needed. The article firstly models three different control types of inverters, then starts from the IEEE 9-bus interconnected system composed of multiple types of inverters, models the system and analyzes its stability by root trajectory calculation, and obtains the influence of each parameter change on the stability of the system, based on this and optimizes the sensitive parameters of the system by sensitivity analysis, so as to enhance the stability of the system. Finally, the theory is verified by time domain simulation, and the correctness of the model is verified.

Study of robust stability of indoor temperature control system

The stages of design and study of the indoor temperature control system have been considered. An assessment of the possibilities of the indoor temperature control of a discrete model of the control system has been carried out. The transfer function of the control object with interval coefficients of the numerator and denominator polynomials taking into account possible ranges of change of the greatest time constant responsible for the enclosing structures has been obtained. The applied graphical method of studying the obtained system has provided an output of the family of root trajectories on a complex plane and visually demonstrated absolute and robust absolute stability of the automatic temperature control system. The simulation carried out in the MATLAB system has confirmed the correctness of the results obtained.

Origin, branching pattern, foraminal and intraspinal distribution of the human lumbar sinuvertebral nerves

BACKGROUND CONTEXTThe lumbar sinuvertebral nerve (SVN) innervates the outer posterior intervertebral disc (IVD); it is thought to mediate discogenic low-back pain (LBP). Controversy, however, exists on its origins at higher (L1–L2) versus lower (L3–L5) lumbar levels. Additionally, lack of knowledge regarding its foraminal and intraspinal branching patterns and extensions may lead to iatrogenic damage. PURPOSETo systematically describe the origins of the L2 and L5 SVNs, their morphological variation in the intervertebral foramen (IVF) and intraspinal distribution. STUDY DESIGNDissection-based study of 20 SVNs with histological confirmation in five embalmed human cadavers. METHODSThe origin, branching pattern and distribution of the L2 and L5 SVNs was investigated bilaterally in five human cadavers using dorsal and anterolateral dissection approaches. Parameters studied included somatic and/or autonomic SVN root contributions, foraminal SVN morphology and course, diameter, branching point, intraspinal distribution and IVD innervation pattern. Nerve tissue was confirmed by immunostaining for neurofilament and S100 proteins. RESULTSThe SVN and its origins was identified in all except one IVF at L2 and in all foramina at L5. At L2, the SVN arose in nearly 90% of sides from both somatic and autonomic roots and at L5 in 40% of sides. The remaining SVNs were formed by purely autonomic roots. The SVN arose from significantly more roots at L2 than L5 (3.1 ± 0.3 vs. 1.9 ± 0.3, respectively; p=.022). Four different SVN morphologies could be discerned in the L2 IVF: single filament (22%), multiple (parallel or diverging) filament (33%), immediate splitting (22%) and plexiform (22%) types, whereas the L5 SVN consisted of single (90%) and multiple (10%) filament types. SVN filaments were significantly thicker at L2 than L5 (0.48 ± 0.06 mm vs. 0.33 ± 0.02 mm, respectively; p=.043). Ascending SVN filaments coursed roughly parallel to the exiting spinal nerve root trajectory at L2 and L5. Branching of the SVN into ascending and descending branches occurred mostly intraspinal both at L2 and L5. Spinal canal distribution was also similar for L2 and L5 SVNs. Lumbar posterior IVDs were innervated by the descending branch of the parent SVN and ascending branch of the subjacent SVN. CONCLUSIONSThe SVN at L2 originates from both somatic and autonomic roots in 90% of cases and at L5 in 40% of cases. The remaining SVNs are purely autonomic. In the IVF, the L2 SVN is morphologically heterogeneous, but generally consists of numerous filaments, whereas at L5 90% contains a single SVN filament. The L2 SVN is formed by more roots and is thicker than the L5 SVN. Intraspinal SVN distribution is confined to its level of origin; lumbar posterior IVDs are innervated by corresponding and subjacent SVNs (ie, two spinal levels). CLINICAL SIGNIFICANCEOur findings indicate that L5 discogenic LBP may be mediated both segmentally and nonsegmentally in 40% of cases and nonsegmentally in 60% of cases. Failure of lower lumbar discogenic pain treatment may be the result of only interrupting the nonsegmental pathway, but not the segmental one as well. Relating SVN anatomy to microsurgical spinal approaches may prevent iatrogenic damage to the SVN and the formation of postsurgical back pain.

Small-signal modeling and stability analysis of grid-connected offshore wind power based on virtual synchronous generator control

With the increasing installed capacity of offshore wind power, the grid-connected of offshore wind power technology based on virtual synchronous generators (VSG) control can simulate the inertia and damping characteristics of synchronous generator, which is helpful to improve the networking ability and inertial supportability of grid-connected inverter. However, VSG control also inevitably causes system oscillation, endangers the safe and stable operation of the power grid. Hence, the main circuit and control circuit of offshore wind power grid connection are firstly modeled, and the small-signal model of offshore wind power grid-connected inverter based on VSG control is obtained. Then the correctness of the model is verified by Matlab/Simulink software. Finally, the root trajectory method is used to identify the effects of VSG control parameters and line parameters on the system stability. The results show that VSG control has a significant influence on the stability of the grid-connected system, and reasonable design parameters are needed to ensure the safe and stable operation of the power grid.

Diverse Dance Synthesis via Keyframes with Transformer Controllers

AbstractExisting keyframe‐based motion synthesis mainly focuses on the generation of cyclic actions or short‐term motion, such as walking, running, and transitions between close postures. However, these methods will significantly degrade the naturalness and diversity of the synthesized motion when dealing with complex and impromptu movements , e.g., dance performance and martial arts. In addition, current research lacks fine‐grained control over the generated motion, which is essential for intelligent human‐computer interaction and animation creation. In this paper, we propose a novel keyframe‐based motion generation network based on multiple constraints, which can achieve diverse dance synthesis via learned knowledge. Specifically, the algorithm is mainly formulated based on the recurrent neural network (RNN) and the Transformer architecture. The backbone of our network is a hierarchical RNN module composed of two long short‐term memory (LSTM) units, in which the first LSTM is utilized to embed the posture information of the historical frames into a latent space, and the second one is employed to predict the human posture for the next frame. Moreover, our framework contains two Transformer‐based controllers, which are used to model the constraints of the root trajectory and the velocity factor respectively, so as to better utilize the temporal context of the frames and achieve fine‐grained motion control. We verify the proposed approach on a dance dataset containing a wide range of contemporary dance. The results of three quantitative analyses validate the superiority of our algorithm. The video and qualitative experimental results demonstrate that the complex motion sequences generated by our algorithm can achieve diverse and smooth motion transitions between keyframes, even for long‐term synthesis.

Coordinated Sending-End Power System Frequency Regulation via UHVDC

The continuous improvement of new energy penetration reduces the inertia of the system, which leads to the frequency deviation and the rate of change of frequency (RoCoF) being easily exceeded. To improve the frequency stability of sending-end power systems with large-scale renewable energy access via ultra-high voltage direct current (UHVDC), the coordinated frequency control for UHVDC participating in system frequency regulation (FR) including primary FR and system inertial response is presented. Based on the simplified system model, the mechanism of UHVDC participation in system frequency support and its influence on receiving-end system frequency response characteristics are analyzed. Compared with the inertia response and primary FR of traditional synchronous generators, the parameter calculating method of UHVDC coordinated frequency response control is proposed. Based on the system root trajectory analysis, the influence of the frequency response control parameters on the sending-end system’s stability is analyzed, and the constraints of UHVDC participating in the system frequency response control are analyzed. Then, based on the RTDS verification platform containing the Lingshao ±800 kV UHVDC control and protection system, the system frequency response characteristics under different control strategies, operating conditions and control parameters are verified and analyzed. The experimental results show that the UHVDC frequency coordinated control can effectively increase the equivalent inertia of the sending-end system, restrain the RoCoF and the frequency deviation, and increase the FR capability of the UHVDC system.