Equilibrium Manifold Research Articles

Dynamic modelling and equilibrium manifold of multi‐converter systems: A study on grid‐forming and grid‐following converters in renewable energy power plants

AbstractThis study explores the optimal balance between grid‐forming (GFM) and grid‐following (GFL) converter capacities within power stations to ensure stable operations. The investigation introduces a novel, generic modelling approach for analysing multiple converter systems in the wind and photovoltaic power plants. The method aims to elucidate the dynamic characteristics of the converters in power plants, particularly focusing on the continuity and existence of the equilibrium manifolds and their impact on system stability. Findings reveal a pronounced difference in the recovery capabilities of GFM and GFL following synchronization losses, highlighting an asymmetry in their abilities. Specifically, GFL converters exhibit more effectiveness in reinstating synchrony after synchronization losses caused by GFM. Conversely, GFM demonstrates a lesser capacity to recover from synchronization losses induced by GFL. Furthermore, analysis indicates that when the capacity ratio of GFL to the system's short‐circuit capacity significantly exceeds that of GFM (exceeding a 1:5 ratio), the system experiences an absence of a stable equilibrium point, thereby affecting the synchronization stability of GFM. These conclusions have been validated through joint controller hardware‐in‐the‐loop testing.

Improved real-time [formula omitted] control for aero-engines based on the equilibrium manifold expansion model

This paper investigates the transient performance and disturbance rejection control problem for aero-engines based on the equilibrium manifold expansion (EME) model. Considering the operating characteristics of the aero-engine vary greatly during different flight states, it is difficult to ensure the performance of the aero-engine in different states using a single controller designed offline. To solve this problem, this paper proposes an improved real-time H∞ controller. Through driving the controller to match the changes in dynamic characteristics of the system, the stability and disturbance rejection performance of the system are guaranteed over a wide range effectively. In addition, a dynamic adjustment mechanism is designed for the disturbance rejection parameter in the proposed real-time H∞ controller, which further improves the transient performance of the system. The sufficient conditions to ensure the stability of the system under the designed method are also given. Finally, the effectiveness and superiority of the proposed method are verified through an aero-engine numerical simulation program and a hardware-in-the-loop (HIL) experiment.

Constrained Model Predictive Control for Generation Power Distribution on Aircraft Engines

Aiming at the increasing demand for electric energy in aircraft in the future, a multi-objective optimization aircraft engine constrained model predictive control method based on generation power distribution is proposed. Firstly, based on the aircraft engine component level model and the equilibrium manifold theory, the aircraft engine equilibrium manifold expansion model is established. Secondly, the influence of the power generation is modeled, and the influence of the low- and high-pressure shaft generators on the normal operation of the aircraft engine is studied and compared. The control variables such as fuel flow and total generation power are taken as the constraint conditions to design the constraint model predictive controller. Furthermore, the multi-objective grey wolf optimization algorithm is introduced to intelligently optimize the parameters of the designed controller. At last, the simulation based on the component level model shows that the high-pressure shaft generator has less influence on the state quantity, including engine thrust, than the low-pressure shaft generator. The proposed control method using the multi-objective gray wolf optimization (MOGWO) algorithm has rapid response and no steady-state error.

A rapid-response soft end effector inspired by the hummingbird beak.

Biology is a wellspring of inspiration in engineering design. This paper delves into the application of elastic instabilities-commonly used in biological systems to facilitate swift movement-as a power-amplification mechanism for soft robots. Specifically, inspired by the nonlinear mechanics of the hummingbird beak-and shedding further light on it-we design, build and test a novel, rapid-response, soft end effector. The hummingbird beak embodies the capacity for swift movement, achieving closure in less than [Formula: see text]. Previous work demonstrated that rapid movement is achieved through snap-through deformations, induced by muscular actuation of the beak's root. Using nonlinear finite element simulations coupled with continuation algorithms, we unveil a representative portion of the equilibrium manifold of the beak-inspired structure. The exploration involves the application of a sequence of rotations as exerted by the hummingbird muscles. Specific emphasis is placed on pinpointing and tailoring the position along the manifold of the saddle-node bifurcation at which the onset of elastic instability triggers dynamic snap-through. We show the critical importance of the intermediate rotation input in the sequence, as it results in the accumulation of elastic energy that is then explosively released as kinetic energy upon snap-through. Informed by our numerical studies, we conduct experimental testing on a prototype end effector fabricated using a compliant material (thermoplastic polyurethane). The experimental results support the trends observed in the numerical simulations and demonstrate the effectiveness of the bio-inspired design. Specifically, we measure the energy transferred by the soft end effector to a pendulum, varying the input levels in the sequence of prescribed rotations. Additionally, we demonstrate a potential robotic application in scenarios demanding explosive action. From a mechanics perspective, our work sheds light on how pre-stress fields can enable swift movement in soft robotic systems with the potential to facilitate high input-to-output energy efficiency.

Observer Design and State-Feedback Stabilization for Nonlinear Systems via Equilibrium Manifold Expansion Linearization

Observer Design and State-Feedback Stabilization for Nonlinear Systems via Equilibrium Manifold Expansion Linearization

Fixed-time anti-disturbance control for nonlinear turbofan engine with impulsive prescribed performance

To analyze the robust control problem of turbofan engine within a large flight envelope, a fixed-time prescribed performance control scheme is proposed for the nonlinear turbofan engine in presence of unknown disturbances and possibly discontinuous reference signals. Firstly, the equilibrium manifold expansion (EME) modeling method is adopted for the turbofan engine to construct an affine nonlinear model. Then, considering the possibly discontinuous reference signals, a novel fixed-time prescribed performance function (FxTPPF) with impulsive characteristic is presented to successfully deal with the singularity problem caused by its discontinuity situation. In what follows, a fixed-time nonlinear disturbance observer (FxTNDO) is developed to realize fast estimation of disturbances within fixed time. According to the transformed error system and disturbance estimations, a fixed-time anti-disturbance composite controller is further developed. By borrowing Lyapunov stability theory, the boundedness of all error signals in the impulsive closed-loop system is proved. Finally, simulation results indicate that this control scheme can ensure rotor speeds effectively to track the desired commands within a predefined time after the appearance of a discontinuity and the tracking errors are limited to the performance bound all along.

Numerical and computational analysis on a dissipative dynamical system: Slow invariant manifold for complex chemical mechanism

This article presents the notion about Slow Invariant Manifold (SIM) and their fundamental role in model reduction techniques (MRTs) for challenges encountered in mechanical engineering within dissipative systems of chemical kinetics. Focusing on the reaction routes of complex mechanisms, we construct and compare primary approximations of the SIM through MRTs, including the Spectral Quasi Equilibrium Manifold (SQEM) and Intrinsic Low Dimensional Manifold (ILDM). These methods effectively transform high-dimensional complex problems into lower dimensions, solving them without compromising crucial information about the complex systems modified for homogeneous reactive systems. Employing the sensitivity analysis by using the MATLAB's toolbox, we present the numerical findings in a tabular format obtained through MRTs. This study provides the understanding about the accessible exploration of numerical solutions, improving insights of the complex variation within the system.

Novel reduction schemes for a dissipative dynamical system: A study on slow invariant manifolds in chemical kinetics

This research explores the intricate concept of the Slow Invariant Manifold (SIM) and its pivotal role in developing model reduction techniques (MRTs) for challenges within dissipative systems in chemical kinetics, specifically in mechanical engineering. Focusing on the multi-step mechanism with two intermediates, primary approximations of the SIM are constructed and compared using two prominent MRTs: The Spectral Quasi Equilibrium Manifold (SQEM) and Intrinsic Low Dimensional Manifold (ILDM). At the given rate coefficient, a special computational experiment was performed in which the efficiency of chemical species has been compared. Noteworthy innovation involves evaluating SIM separately for reduced species, departing from the conventional approach of considering every species within the mechanism. The study employs local sensitivity analysis with MATLAB's Sim-Biology toolbox, presenting quantitative findings in a tabular format for a comprehensive MRT comparison. Beyond contributing to a deeper understanding of model reduction techniques in complex chemical kinetics, this research marks the first computational exploration of dissipative systems. The novel perspective of evaluating SIM for reduced species offers nuanced insights, emphasizing the critical role of SIM in effectively addressing challenges in mechanical engineering applications. In summary, this study introduces computational advancements and novel approaches, advancing model reduction techniques and highlighting the significance of SIM in addressing challenges within mechanical engineering contexts.

Analyzing the spatial motion of a rigid body subjected to constant body-fixed torques and gyrostatic moment

This paper aims to explore the rotatory spatial motion of an asymmetric rigid body (RB) under constant body-fixed torques and a nonzero first component gyrostatic moment vector (GM). Euler's equations of motion are used to derive a set of dimensionless equations of motion, which are then proposed for the stability analysis of equilibrium points. Specifically, this study develops 3D phase space trajectories for three distinct scenarios; two of them are applied constant torques that are directed on the minor and major axes, while the third one is the action of applied constant torque on the body’s middle axis. Novel analytical and simulation results for both scenarios of constant torque applied along the minor and middle axes are provided in the context of separatrix surfaces, equilibrium manifolds, periodic or non-periodic solutions, and periodic solutions’ extreme. Concerning the scenario of a directed torque on the major axis, a numerical solution for the problem is presented in addition to a simulation of the graphed results for the angular velocities' trajectories in various regions. Moreover, the influence of GM is examined for each case and a full modeling for the body's stability has been present. The exceptional impact of these results is evident in the development and assessment of systems involving asymmetric RBs, such as satellites and spacecraft. It may serve as a motivating factor to explore different angles within the GM in similar cases, thereby influencing various industries, including engineering and astrophysics applications.

Inertial migration of spherical and oblate particles in a triangular microchannel.

The. inertial migration of both spherical and oblate particles within an equilateral triangular channel is studied numerically. Our study primarily focuses on the effects of fluid inertia, quantified by the Reynolds number (Re) and particle size (β). Our observations reveal two distinct equilibrium positions: the corner equilibrium position (CEP) is situated along the angle bisector near the corner, while the face equilibrium position (FEP) is located on a segment of the line perpendicular from the triangle's center to one of its sides. Spherical particles with varying initial positions predominantly reach the FEP. For oblate particles initially positioned along the angle bisector with a specific orientation, meaning the particle's evolution axis is inside the plane bisecting the angle, they will migrate along the angle bisector to reach the CEP while rotating in the tumbling mode. Conversely, for particles with different initial orientations and positions, they will employ the log-rolling mode to reach the FEP. Notably, we identify a dual-stage particle migration process to the FEP, with trajectories converging to an equilibrium manifold, which bears a resemblance to the cross sectionof the channel. To further illustrate the transition between FEP and CEP under general initial conditions, except for those along the angle bisector, we construct a phase diagram in the (Re, β) parameter space. This transition is often triggered by the size of larger particles (as the FEP cannot accommodate them) or the influence of inertia for smaller particles. For the FEP, especially for medium- or small-size particles, we notice an initial outward movement of the FEP from the center of the cross sectionas Re increases, followed by a return towards the center. This behavior results from the interplay of three forces acting on the particle. This research holds potential implications for the design of microfluidic devices, offering insights into the behavior of particles within equilateral triangular channels.

Multi-Parameter Practical Stability Region Analysis of Wind Power System Based on Limit Cycle Amplitude Tracing

The existing division of system stability region does not fully consider the influence of nonlinear characteristics of the system. To address the challenge that it is difficult to inscribe the stability region of wind power grid-connected systems accurately and quantitatively, this paper proposes a nonlinear system multi-parameter practical stability region analysis method based on limit cycle amplitude tracking. Firstly, the nonlinear dynamic model of multi-machine system with doubly fed induction generator (DFIG) is established, and the bifurcation diagram of the system is obtained by nonlinear analysis of the model. Then, the limit cycle-parameter diagram is obtained by tracking the limit cycle of the Hopf bifurcation point of the system, and the problem of how the system oscillation evolves with the change of operating parameters is analyzed. Afterwards, the single parameter practical stability region of the nonlinear system is obtained by combining the system equilibrium manifold. Finally, the multi-parameter practical stability region of the nonlinear system is obtained by the three-dimensional bifurcation map with the system time delay, DFIG output active power and reactive load as the bifurcation parameters. The division of the stability region has important guiding significance for the parameter adjustment in the actual operation of the system.

Intelligent ammonia precooling control for TBCC mode transition based on neural network improved equilibrium manifold expansion model

Precise control of the aeroengine precooling temperature is important for safeguarding the mode transition (MT) in turbine-based combined cycle (TBCC) propulsion. Ammonia mass injection pre-compressor cooling (Ammonia MIPCC) technology increases the maximum operating Mach number of aeroengines, presenting challenges in the design of controllers for complex control objectives. This paper introduces the combined inlet characteristics and the precooling scheme in the MT and proposes a neural network-improved equilibrium manifold expansion (NNEME) model. An intelligent ammonia precooling controller for the ammonia MIPCC aeroengine (AMA) was established based on NNEME and extended state observer (ESO). Furthermore, the final error confidence intervals for the NNEME model show a significant reduction of 37.499 % for rotor speed and 37.919 % for compressor outlet temperature, compared to the conventional EME model. By implementing ESO-based online compensation for modeling uncertainties, we achieved remarkable improvements in the control results. Specifically, the maximum steady-state errors for rotor speed and compressor outlet temperature were reduced by 60.839 % and 51.529 %, respectively, when compared to control results without ESO integration. Therefore, the proposed NNEME + ESO-based controller enabled precise control of the AMA precooling temperature during the MT phase. This demonstrates the effectiveness of our approach in enhancing control accuracy and stability.

Super-twisting sliding mode finite time control of power-line inspection robot with external disturbances and input delays

An adaptive super-twisting sliding mode control (AST-SMC) is proposed for the power-line inspection (PLI) robot system with two degrees of freedom and single control input underactuated with multiple balances. Firstly, the nonlinear dynamic equation of PLI robot is established, and the model is linearized at the nominal equilibrium point. Secondly, a special transformation is introduced to transform the original input delay system into a delay-free model, and an equilibrium manifold linearization (EML) model is established by using scheduling variables to expand the operating range of the controlled system, as a result of which the disturbance observer control (DOC) is designed to estimate the external disturbance. Thirdly, in order to achieve fast convergence and continuous control, the super-twisting sliding mode control(ST-SMC) method is proposed to decrease chattering in the control law. Finally, it can be verified by Lyapunov stability theory that the system reaches the sliding mode surface in finite time and the trajectory tracking error converges to zero in finite time. The simulation results verify that the combination of DOC and AST-SMC has a optimal disturbance approximation and control convergence ability, clarifying the effectiveness and robustness of the proposed control scheme.

Stability analysis of the Chua’s circuit with generic odd nonlinearity

This paper analyzes the stability of the Chua’s circuit with generic odd nonlinearity using two traditional analytical tools of the control theory. The first tool is based on the linear approximation of a nonlinear system for the local stability analysis around equilibrium points or manifolds using the Jacobian matrix eigenvalues, which are mapped in the complex plane using the root locus method. The second tool is an extension of linear techniques based on frequency response known as describing function method, which allows analyze effects of nonlinearities in dynamical systems and predict several nonlinear phenomena with reasonable accuracy. These two analytical tools are jointly applied to the stability analysis of an example of this class of nonlinear systems to identify and map its dynamical behavior in parameter space. Numerical investigations based on computational simulations corroborate the theoretical predictions obtained in this stability analysis.

Gyroscopic Balancer-Enhanced Motion Control of an Autonomous Bikebot

Abstract Bikebot (i.e., bicycle-based robot) is a class of underactuated balance robotic systems that require simultaneous trajectory tracking and balance control tasks. We present a tracking and balance control design of an autonomous bikebot. The external-internal convertible structure of the bikebot dynamics is used to design a causal feedback control to achieve both the tracking and balance tasks. A balance equilibrium manifold is used to define and capture the platform balance profiles and coupled interaction with the trajectory tracking performance. To achieve fully autonomous navigation, a gyrobalancer actuation is integrated with the steering and velocity control for stationary platform balance and stationary-moving switching. Stability and convergence analyses are presented to guarantee the control performance. Extensive experiments are presented to validate and demonstrate the autonomous control design. We also compare the autonomous control performance with human riding experiments and similar action strategies are found between them.

Equilibrium selection under changes in endowments: A geometric approach

In this paper we propose a geometric approach to the selection of the equilibrium price. After a perturbation of the parameters, the new price is selected thorough the composition of two maps: the projection on the linearization of the equilibrium manifold, a method that underlies econometric modeling, and the exponential map, that associates a tangent vector with a geodesic on the manifold. As a corollary of our main result, we prove the equivalence between zero curvature and uniqueness of equilibrium in the case of an arbitrary number of goods and two consumers, thus extending the previous result by Loi and Matta (2018).

Stability analysis of an acted asymmetric rigid body by a gyrostatic moment and a constant body-fixed torque

This work focuses on the dynamical rotatory motion of an asymmetric rigid body (RB) subjected to a constant body-fixed torque and a vector of a gyrostatic moment (GM). The motion is considered in the absence of the first two components of the GM. The novelty of the current work is the conversion of the stability analysis of this body from a two dimensional phase plane to three ones phase space. Specifically, two scenarios of stationary torques are considered, the first one is directed on the minor or major axis, while the latter is presumed to act on the middle axis. Therefore, three dimensional phase space routes are generated. In both scenarios some novel analytical and simulation outcomes are provided regarding equilibrium manifolds, periodic solutions or non-periodic ones, separator surfaces, as well as the extreme periodic solutions. The significance of this work is due to its great applications, especially those that use the gyro’s theory.

The problem of reduce description in chemical kinetics

A comprehensive comprehension of the concept of the gradual unchanging manifold (GUM) gives rise to numerous techniques for reducing models (TRM) in dissipative systems of chemical kinetics for mechanical engineering problems. This research calculates and contrasts the unchanging solutions for multi-route systems. The TRM, such as the Intrinsic Low Dimensional (ILDM), Quasi Equilibrium Manifold (QEM), and Spectral Quasi Equilibrium Manifold (SQEM), are utilized to construct the gradual unchanging manifolds, which function as primary approximations to GUM. By employing the Method of Invariant Grid (MIG), the enhancements of QEM solutions are computed imposing the effect of activation energy. Our discoveries suggest that we could assess each route independently rather than taking into account the entire mechanism. Local sensitivity analysis is conducted using the SimBiology toolbox of MATLAB. To compare TRM, the calculation period is presented in tabular form.

Extended-Kalman-filter-based equilibrium manifold expansion observer for ramjet nonlinear control

A high-accuracy control-oriented model is an important prerequisite for the design of a ramjet controller, which can provide good engine performance and ensure safe and reliable operation under various environments and working conditions. In this study, we established a control-oriented equilibrium manifold expansion (EME) model for ramjets based on engine operation data. Furthermore, according to the EME model, an uncertainty term was introduced to reflect the steady-state error of the system. Meanwhile, in combination with the extended Kalman filter (EKF) algorithm, an EME observer was developed to eliminate the steady-state error and amend the model. Then, we established a model that can reflect the maneuvering characteristics of a ramjet in a large envelope range. Moreover, a feasible nonlinear controller was designed using the EME model and the sliding mode control (SMC) method to further eliminate the influence of model parameter perturbations and enhance the robustness of the control system. A direct control target was designed to further verify the estimation accuracy of the unmeasurable output of the EKF-based EME observer, and to achieve the direct control of the thrust performance and safety performance of a ramjet. Finally, the results of the hardware-in-the-loop (HIL) experiment showed that the state variable tracking error of the EME + EKF SMC system was reduced by at least 40% under continuous variable Mach number instructions, and the tracking error was reduced by more than 70% under more extreme simulation instructions. The control system was robust, and the accuracy of the model significantly improved. In addition, the results of the HIL experiment showed that a EME + EKF SMC control system with high accuracy and stability was established with thrust and inlet stability margin as control objectives.

Control of a jet engine using predictor-based adaptive strategy in different flight modes

In this article, a predictor-based model reference adaptive control method is proposed for a JetCat SPT5 turboshaft engine in full thrust, cruise, and idle modes. In the predictor-based model reference adaptive control method, in addition to the tracking error, the predictor error is also utilized in adaptive laws to achieve the desired control objectives, such as improving the tracking performance and control signals, and reducing the tracking error. The proposed method is implemented on a turboshaft engine system with actual dynamic values. First, three properly separated equilibrium points are selected on the nominal plant equilibrium manifold to linearize the plant model at the equilibrium points. Then, the predictor-based model reference adaptive control controller is designed and investigated for the three equilibrium points generating three operating modes of the engine, that is, full thrust, cruise, and idle. The stability of the control system is proved by the Lyapunov method. To evaluate the efficiency of the state predictor method in the simulation scenarios, the proposed method is compared with the classic model reference adaptive control method. The simulation results illustrate the superiority of the predictor-based model reference method due to the reduction of unwanted fluctuations in tracking performance, faster convergence of the tracking error to zero, and smoother control signals for all the equilibrium points.