Theoretical Stability Analysis Research Articles

Stability and control of power grids with diluted network topology.

We consider sparse random networks of Kuramoto phase oscillators with inertia in order to mimic and investigate the dynamics emerging in high-voltage power grids. The corresponding natural frequencies are assumed to be bimodally Gaussian distributed, thus modeling the distribution of both power generators and consumers, which must be in balance. Our main focus is on the theoretical analysis of the linear stability of the frequency-synchronized state, which is necessary for the stable operation of power grids and the control of unstable synchronous states. We demonstrate by numerical simulations that unstable frequency-synchronized states can be stabilized by feedback control. Further, we extend our study to include stochastic temporal power fluctuations and discuss the interplay of topological disorder and Gaussian white noise for various model configurations and finally demonstrate that our control scheme also works well under the influence of noise. Results for synthetic Erdös-Renyi random networks with low average connectivity and with symmetric or asymmetric bimodal frequency distributions are compared with those obtained by considering a real power grid topology, namely, the grid of Italy.

Concurrent Learning Adaptive Control With Directional Forgetting

This paper proposes a new concurrent learning-based adaptive control algorithm. The main objective behind our proposition is to relax the persistent excitation requirement for the stability guarantee, while providing the ability to identify time-varying parameters. To achieve the objective, this paper designs a directional forgetting algorithm, which is then integrated with the adaptive law. The theoretical stability analysis shows that the tracking and parameter estimation error is exponentially stable with the signal only finitely excited, not persistently excited. The analysis also shows that the proposed algorithm can guarantee the stability under time-varying parameters. Moreover, the necessary and sufficient conditions for the stability given the time-varying parameters are derived. The results of numerical simulations confirm the validity of the theoretical analysis results and demonstrate the performance of the proposed algorithm.

On Model-Free Adaptive Control and Its Stability Analysis

In this paper, the main issues of model-based control methods are first reviewed, followed by the motivations and the state of the art of the model-free adaptive control (MFAC). MFAC is a novel data-driven control method for a class of unknown nonaffine nonlinear discrete-time systems since neither explicit physical model nor Lyapunov stability theory or key technical lemma is used in the controller design and theoretical analysis except only for the input/output (I/O) measurement data. The basis of MFAC is the dynamic linearization data modeling method at each operating point of the closed-loop system. The established dynamic linearization data model is a virtual equivalent data relationship in the I/O sense to the original nonlinear plant by means of a novel concept called pseudo-partial derivative (PPD) or pseudo-gradient (PG) vector. Based on this virtual equivalent dynamic linearization data model and the time-varying PPD or PG estimation algorithm designed merely using the I/O measurements of a controlled plant, the MFAC system is constructed. The main contribution of this paper is that the theoretical analysis of the bounded-input bounded-output stability, the monotonic convergence of the tracking error dynamics, and the internal stability of the full form dynamic linearization based MFAC scheme are rigorously presented by the contraction mapping principle; the well known PID control and the traditional adaptive control for linear time-invariant systems are explicitly shown as the special cases of this MFAC. The simulation results verify the effectiveness of the proposed approach.

Meso-structural characteristics of asphalt mixture main skeleton based on meso-scale analysis

To provide a reference for the gradation design of dense skeleton asphalt mixture (DSAM), a meso-scale main skeleton structural system with a contact chain as the medium scale, composed of coarse aggregates and contact points, contact chains, and a main skeleton, is proposed in this study. Based on the theoretical analysis of the main contact chain structure stability, the statistical characteristics of the main skeleton, contact chain of the asphalt mixtures, as well as their gradual behaviour under loading, were studied using the digital image processing (DIP) techniques. At last, the corresponding meso-evaluation indices and design criteria for the main skeleton were proposed. The results indicate that the factors affecting the stability of the main contact chain include the particle size of the coarse aggregate, the length, structure, and direction of the main contact chain, and the cohesion of asphalt binder. Coarser gradation and a shorter contact chain, as well as a linear main contact chain with the same load action direction, exhibit greater stiffness and superior resistance to deformation. A lower number of contact chains and longer contact chains in the asphalt mixture result in a more perfect main skeleton. When creating asphalt mixture specimens using the wheel rolling method, the design criteria of the main skeleton structure meso-index should reach Ks ≥ 80% and Nn ≤ 37. The design criteria can be used as a reference basis for the design of DSAM with a nominal maximum particle size of 13.2 mm, ensuring that the coarse aggregates are interlocked to form a stable main skeleton.

Bacterial spinning top

We have investigated the dynamics of a monotrichous bacteria cell near a wall boundary, taking elastic hook flexibility into consideration. Combining theoretical linear stability analysis and direct numerical computations via the boundary element method, we have found that the elastohydrodynamic coupling between the hook elasticity and cell rotational motion enables a stable vertical spinning behaviour like a low-Reynolds-number spinning top. The forwardly rotated flagellum, which generates the force exertion pushing towards the cell body, typically destabilizes the vertical upright position and leads to a boundary-following motion. In contrast, the backward rotation of the flagellum, generating a force pulling the cell body, contributes to stable upright behaviour in a large range of hook rigidity. Further numerical investigations have demonstrated that the non-spherical geometry of the cell body and boundary adhesive interactions affect the bacterial dynamics, leading to complex behaviours such as horizontal spinning and unstable vertical spinning motions, both of which are experimentally observed in Pseudomonas aeruginosa bacteria. These results highlight the rich diversity of bacterial surface motility emerging from mechanical boundary interactions coupled with the cell swimming and hook flexibility.

Robust tracking control for vehicle electronic throttle using adaptive dynamic sliding mode and extended state observer

• The mathematical modelling of automotive engine electronic throttle (EET) system is obtained by considering the nominal model and parameter uncertainty component and all the system uncertainties are detailedly analyzed, then a modified extended state observer (ESO) is adopted to estimate the lumped uncertainty online. • An adaptive dynamic sliding mode (ADSM) feedback control scheme with ESO techniques is designed to achieve a robust closed-loop tracking performance. By adopting the ESO and designing the dynamic secondary sliding surface (DSSS), the chattering phenomenon of the conventional sliding mode can be significantly alleviated. • The additional dynamics introduced in the sliding mode can improve the system stability. A detailed theoretical stability analysis for the whole closed-loop EET control system is presented by innovatively combining the observer dynamics with the feedback control system. • The real-time experimental results from the DSP-based EET experimental platform are given with performance comparisons with other controllers, and both the excellent performance and robustness of the proposed control are achieved. • The robust adaptive dynamic sliding mode control scheme for EET systems will provide the automotive engineers with a good candidate of robust EET controller in practical applications. The proposed control technique can also be applied to other automotive systems for achieving excellent control performance. This article proposes an extended state observer (ESO)-based robust adaptive dynamic sliding mode (ADSM) tracking control for an automobile engine electronic throttle (EET) system. The newly developed control scheme exhibits the following three attractive characteristics. First, a modified ESO is adopted to estimate the lumped uncertainty online. Second, an ADSM feedback control is then designed to achieve a robust closed-loop tracking performance. Third, the switching gain is adaptively updated by the adaptive law to eliminate the need of the uncertainty information. The stability proof combining the observer with the feedback control is presented in detail. High tracking precision and robustness of the proposed control are verified by extensive experimental results with comparison of existing adaptive sliding mode control and conventional robust controllers.

Multi-guide particle swarm optimization for multi-objective optimization: empirical and stability analysis

This article presents a new particle swarm optimization (PSO)-based multi-objective optimization algorithm, named multi-guide particle swarm optimization (MGPSO). The MGPSO is a multi-swarm approach, where each subswarm optimizes one of the objectives. An archive guide is added to the velocity update equation to facilitate convergence to a Pareto front of non-dominated solutions. An extensive empirical and stability analysis of the MGPSO is conducted. The empirical analysis focuses on the exploration behavior of the MGPSO and compares the performance of the MGPSO with that of state-of-the-art multi-objective PSO and evolutionary algorithms. The results show that the MGPSO is highly competitive on a number of benchmark functions. The paper provides a theoretical stability analysis which focuses on the sufficient and necessary conditions for order-1 and order-2 stability of the MGPSO. The paper extends existing work on MGPSO stability analysis by deriving new stability criteria for differing values of the acceleration coefficients used in the velocity update equation.

A pseudospectral method based robust-optimal attitude control strategy for spacecraft

This paper proposes an optimal integral sliding mode control (ISMC) scheme for attitude regulation of the rigid spacecraft. This control technique is capable of handling inertial matrix uncertainties as well as external disturbances. To incorporate optimality into the robust control law, the ISMC is integrated with Legendre pseudospectral method (LPSM). The minimization of the cost function and constraint handling of the spacecraft is obtained by LPSM, while the ISMC provides disturbance rejection. LPSM is chosen for its relatively high rate of convergence and its capability of solving a wide range of challenging optimal control problems. Theoretical stability analysis of closed loop system using Lyapunov theorem guarantees the convergence of attitude states. A comparative analysis between the proposed LPSM-ISMC and Chebyshev Pseudospectral Method (CPSM) based ISMC, is also presented in this paper. The effectiveness of the proposed robust-optimal control strategy is established through simulation results.

Piezoelectric Shunt Stiffness in Rhombic Piezoelectric Stack Transducer with Hybrid Negative-Impedance Shunts: Theoretical Modeling and Stability Analysis.

Negative-capacitance shunted piezoelectric polymer was investigated in depth due to its considerable damping effect. This paper discusses the novel controlled stiffness performance from a rhombic piezoelectric stack transducer with three hybrid negative-impedance shunts, namely, negative capacitance in series with resistance, negative capacitance in parallel with resistance, and negative inductance/negative capacitance (NINC) in series with resistance. An analytical framework for establishing the model of the coupled system is presented. Piezoelectric shunt stiffness (PSS) and piezoelectric shunt damping (PSD) are proposed to analyze the stiffness and damping performances of the hybrid shunts. Theoretical analysis proves that the PSS can produce both positive and negative stiffness by changing the negative capacitance and adjustable resistance. The Routh–Hurwitz criterion and the root locus method are utilized to judge the stability of the three hybrid shunts. The results point out that the negative capacitance should be selected carefully to sustain the stability and to achieve the negative stiffness effect of the transducer. Furthermore, negative capacitance in parallel with resistance has a considerably better stiffness bandwidth and damping performance than the other two shunts. This study demonstrates a novel electrically controlled stiffness method for vibration control engineering.

Divided DQ Small-Signal Model: A New Perspective for the Stability Analysis of Three-Phase Grid-Tied Inverters

The instability issues of the three-phase grid-tied inverters are major concerns in the distributed generated systems. However, due to the coupling effects in dq frame, the instability mechanisms of the three-phase inverters have not yet been clearly revealed. This paper presents a divided dq small-signal model of the grid-tied inverter in dq frame. The specific expressions of a small-signal output admittance of the inverter are investigated based on the proposed model, which can reveal the influencing mechanism of coupling effects and phase-locked loop (PLL) on the stability of the inverter. Then, a stability analysis method based on the divided dq small-signal model is proposed, by which the procedure of checking the stability of the grid-tied inverters with strong/weak grid is executed step by step, and it could be helpful for the system parameters design. Furthermore, the stability of the multiparalleled inverter system is also studied, and it indicates that as the number of the paralleled inverters increases, the parameters of PLL could be adjusted for a stable operation. The theoretical stability analysis results are verified by simulations and experiments.

Experimental investigation of theoretical stability regions for ultra-low frequency oscillations of hydropower generating systems

With the increasing momentum towards flexible power systems based on renewables, the role of hydropower has great importance, especially for providing balancing power. In this paper, a fundamental study on the operating stability of hydropower generating systems is conducted to reveal the practical characteristics for the newly emergent issue of ultra-low frequency oscillations. A unique study methodology is adapted by combing the theoretical analysis and the physical model experiment. In this paper, first, the set-up of the integral experiment platform for the transient processes of the pumped storage plants is presented. Second, a mathematical model of hydropower generating systems is built, and the theoretical stability analysis is conducted based on the Routh-Hurwitz criterion and the stability margin region. The model experiments related to the frequency stability of hydropower generating systems were conducted with reference to the stability region from theoretical analysis. The results demonstrate the sustained ultra-low frequency oscillations and frequency instability of hydropower units in experiments for the first time. Attenuation characteristics of the oscillations are theoretically derived based on the stability margin region, and then quantitatively identified by experiments. The experiment accorded with theoretical stability region within a reasonable tolerance that corresponded to the ±0.1 stability margin.

Optimal synchronous vibration control for magnetically suspended centrifugal compressor

Abstract The rotor mass imbalance is common in rotating machinery, which can lead to the synchronous vibration in the magnetically suspended centrifugal compressor (MSCC) system. This work proposes a synchronous vibration control method based on the optimal notch filter. Firstly, the dynamics of the magnetically suspended rotor (MSR) system with mass imbalance is modeled. Then phase-shift notch filter for the synchronous vibration suppression is designed according to theoretical stability analysis of the whole system. Furthermore, in order to optimize the dynamic performance of the notch filter, its phase angle is taken as the variable to be optimized. Taking the integral of the absolute value of time multiplies current as the objective function, the optimized design flow of the phase angle is given. The proposed optimized method does not rely on accurate mathematical model, and is more direct. Finally, the simulation and experiment results demonstrate the effectiveness of the optimal notch filter and also prove the system stability.

A unified approach to fixed-wing aircraft path following guidance and control

This paper addresses the path following control problem for scale-model fixed-wing aircraft. Kinematic guidance and dynamic control laws are developed within a single coherent framework that exploits a simple generic model of aerodynamic forces acting on the aircraft and applies to almost all regular 3D paths. The proposed control solutions are derived on the basis of theoretical stability and convergence analyses, and validated via realistic hardware-in-the-loop simulations.

Stability Analysis of Dual-rate Haptics Controller Using Two Control Architectures

Digital implementation of uniform-rate haptics controller limits the range of virtual environment parameters that can be stably implemented, particularly at higher sampling rates. Dual-rate sampling scheme alleviates this limitation and enlarges the range of these parameters that can be stably implemented. In this paper, theoretical stability analysis of the dual-rate haptics controller is presented using two control architectures. Analytical conditions that guarantee the closed-loop stability of the system are developed using linear control theory tools. The stability criteria constrains the virtual environment parameters as a function of the ratio of sampling periods of the controller, and thereby establish limits on the performance of these controllers. The developed criteria can be seen as the generalization of existing criterion available in the literature.

An efficient space-time operator-splitting method for high-dimensional vector-valued Allen–Cahn equations

PurposeThe purpose of this paper is to propose an efficient space-time operator-splitting method for the high-dimensional vector-valued Allen–Cahn (AC) equations. The key of the space-time operator-splitting is to devide the complex partial differential equations into simple heat equations and nolinear ordinary differential equations.Design/methodology/approachEach component of high-dimensional heat equations is split into a series of one-dimensional heat equations in different spatial directions. The nonlinear ordinary differential equations are solved by a stabilized semi-implicit scheme to preserve the upper bound of the solution. The algorithm greatly reduces the computational complexity and storage requirement.FindingsThe theoretical analyses of stability in terms of upper bound preservation and mass conservation are shown. The numerical results of phase separation, evolution of the total free energy and total mass conservation show the effectiveness and accuracy of the space-time operator-splitting method.Practical implicationsExtensive 2D/3D numerical tests demonstrated the efficacy and accuracy of the proposed method.Originality/valueThe space-time operator-splitting method reduces the complexity of the problem and reduces the storage space by turning the high-dimensional problem into a series of 1D problems. We give the theoretical analyses of upper bound preservation and mass conservation for the proposed method.

Görtler Vortices in Hypersonic Flow on Compression Ramps

Direct numerical simulation of Görtler vortices in hypersonic compression ramp flow is performed. Three ramp angles of 15, 20, and 25 deg are considered for a freestream Mach number of 7.7 and a unit Reynolds number of . The results show that a mesh resolution of 151 million points is sufficient to capture the flow details. The validation tests present excellent agreement with theoretical values and experimental data. Görtler vortices show unsteady features for the considered ramp flows. The induced heat flux variation and the vortex structure of Görtler vortices are discussed. Two methods of determining the Görtler number are proposed. The first one gives the variation of the Görtler number near reattachment based on the computational results. Additionally, based on a simplified model, a formula depending only on the flow condition and the geometry of the ramp model is deduced to predict the Görtler number. It is shown that the latter method can quantitatively predict the Görtler number near reattachment, and the predictions are in qualitative agreement with a theoretical stability analysis. In the flowfield, corrugated reattachment resulting from the three-dimensionality of the separation bubble is found. Görtler vortices originating in the shear layer as a result of streamline curvature are enhanced at and after flow reattachment.

Thermal stability and entropy generation of unsteady reactive hydromagnetic Powell-Eyring fluid with variable electrical and thermal conductivities

Theoretical analysis of thermal stability and inherent irreversibility of reactive hydromagnetic Powell-Eyring fluid flow in a fixed vertical channel with variable electrical and thermal conductivities is examined. The conducting reactive fluid is influenced by gravity and axial pressure gradient under bimolecular kinetic rate law. Ignoring the material assumptions, the governing dimensionless equations are solved by finite semi-discretization difference scheme along with integration Fehlberg Runge-Kutta method. The results for the momentum and heat equations, Bejan number, thermal runaway, entropy generation and current density are graphically presented under different thermophysical parameters. The computational results for the shear stress and heat transfer at the wall is also presented and discussed quantitatively. It was observed that the parameter which enhance the rate of entropy generation also augment Bejan number with large pitch ratio. Hence, entropy generation can be minimized at low dissipation and material variables.

Theoretical and numerical analyses of the structural stability of the pipe-grout-liner system with a crown void subjected to external pressure

The thin-walled polymer liner is widely used to rehabilitate the deteriorated underground pipe. Although the liner is installed close-fitting in the lining procedure, an annular shrinkage gap is inevitable between the liner and the pipe. The grouting technique is commonly employed to eliminate the small gap and often results in a small crown void between the liner and the pipe. Therefore, a pipe-grout-liner system generates. This paper focuses on the structural stability of the pipe-grout-liner system with a crown void subjected to external pressure. It is found that the radially-inward out-of-roundness imperfection results in a single-lobe deformation and the radially-outward out-of-roundness imperfection results in a double-lobe deformation, respectively. The critical buckling pressure is derived and obtained theoretically, and then verified by developing a two-dimensional finite element model. The numerical result shows very close agreement with the analytical solution and other closed-form expressions for both single-lobe and double-lobe cases, respectively. Finally, the effect of the crown void is examined. It is found that the system with a small crown void can sustain the same buckling pressure as the perfectly-confined liner without void. However, a significant reduction of the critical buckling pressure was observed when the crown void is larger than the small size level.

Policy Lag and Sustained Growth

This study provides a theoretical analysis of economic stability when a fiscal policy lag and sustained economic growth coexist. We develop a New Keynesian model by introducing economic growth based on an exogenous factor and a fiscal policy rule with a delay in policy reaction. We clarify the effect of these factors on economic stability by performing a dynamic analysis of the delay-differential equation system derived from the developed model. Our study has several contributions. First, this study contributes an analysis of the effect of a policy lag on stability. When the economy does not grow, a large policy lag destabilizes the economy. However, in a sustainably growing economy, a large policy lag may eliminate instability and contribute to economic stability. Second, we show that a change in the economic growth rate can affect economic stability under a policy lag. This new finding runs contrary to the usual results from growth models without a policy lag.

Analysis of small-signal stability of power systems with photovoltaic generators

This paper presents a theoretical analysis of the small-signal stability of a power system in which a synchronous generator and a photovoltaic (PV) generator supply power to an infinite bus. The problem considered here is to investigate the existence of the equilibrium points of the system and their stability. In terms of this problem, by focusing on the condition to be satisfied by the equilibrium points and appropriately using the intermediate value theorem, we derive a sufficient condition on the magnitude of the PV current for the existence of the equilibrium points. The condition is given as inequalities with respect to the system parameters. These inequalities show that, if the power system from which the PV generator is removed has equilibrium points, then equilibrium points exist also in the original system as long as the PV current is small. Moreover, we analyze the stability of the equilibrium points and show that the equilibrium points found under our existence condition are asymptotically stable. These results imply that, when the PV current is below a certain level, the existence of the asymptotically stable equilibrium points is preserved even though the PV generator is introduced.