Nonlinear Lyapunov Research Articles

Comparison of nonlinear local Lyapunov vectors with bred vectors, random perturbations and ensemble transform Kalman filter strategies in a barotropic model

The breeding method has been widely used to generate ensemble perturbations in ensemble forecasting due to its simple concept and low computational cost. This method produces the fastest growing perturbation modes to catch the growing components in analysis errors. However, the bred vectors (BVs) are evolved on the same dynamical flow, which may increase the dependence of perturbations. In contrast, the nonlinear local Lyapunov vector (NLLV) scheme generates flow-dependent perturbations as in the breeding method, but regularly conducts the Gram–Schmidt reorthonormalization processes on the perturbations. The resulting NLLVs span the fast-growing perturbation subspace efficiently, and thus may grasp more components in analysis errors than the BVs.

On the Ψ-Conditional Exponential Asymptotic Stability of Nonlinear Lyapunov Matrix Differential Equations

Abstract It is proved (necessary and) sufficient conditions for Ψ– conditional exponential asymptotic stability of the trivial solution of nonlinear Lyapunov matrix differential equations

Absolute exponential L1-gain analysis and synthesis of switched nonlinear positive systems with time-varying delay

This paper investigates absolute exponential stability and stabilization with L1-gain performance of switched nonlinear positive systems with time-varying delay. Nonlinear functions considered in the paper are located in a sector field and the time-varying delay is unknown but bounded. The notion of absolute exponential L1 stability of switched nonlinear positive systems is first introduced. A nonlinear Lyapunov–Krasovskii functional is constructed for the underlying systems. By using the Lyapunov–Krasovskii functional, a sufficient condition for absolute exponential L1 stability of the systems is established in terms of linear programming. Then, absolute exponential L1 control synthesis of the systems is addressed. A feedback control law containing nonlinear functions and a state-feedback control law are designed, respectively. By comparing with existing results, it is shown that the present design is less conservative. Finally, an illustrative example is provided to illustrate the effectiveness of the results.

Laminar and turbulent flows over hydrophobic surfaces with shear-dependent slip length

Motivated by extensive discussion in the literature, by experimental evidence and by recent direct numerical simulations, we study flows over hydrophobic surfaces with shear-dependent slip lengths and we report their drag-reduction properties. The laminar channel-flow and pipe-flow solutions are derived and the effects of hydrophobicity are quantified by the decrease of the streamwise pressure gradient for constant mass flow rate and by the increase of the mass flow rate for constant streamwise pressure gradient. The nonlinear Lyapunov stability analysis, first applied to a two-dimensional channel flow by Balogh et al. [“Stability enhancement by boundary control in 2-D channel flow,” IEEE Trans. Autom. Control 46, 1696-1711 (2001)], is employed on the three-dimensional channel flow with walls featuring shear-dependent slip lengths. The feedback law extracted through the stability analysis is recognized for the first time to coincide with the slip-length model used to represent the hydrophobic surfaces, thereby providing a precise physical interpretation for the feedback law advanced by Balogh et al. The theoretical framework by Fukagata et al. [“A theoretical prediction of friction drag reduction in turbulent flow by superhydrophobic surfaces,” Phys. Fluids 18, 051703 (2006)] is employed to model the drag-reduction effect engendered by the shear-dependent slip-length surfaces and the theoretical drag-reduction values are in very good agreement with our direct numerical simulation data. The turbulent drag reduction is measured as a function of the hydrophobic-surface parameters and is found to be a function of the time- and space-averaged slip length, irrespective of the local and instantaneous slip behaviour at the wall. For slip parameters and flow conditions that could be realized in the laboratory, the maximum computed turbulent drag reduction is 50% and the drag reduction effect degrades when slip along the spanwise direction is considered. The power spent by the turbulent flow on the hydrophobic walls is computed for the first time and is found to be a non-negligible portion of the power saved through drag reduction, thereby recognizing the hydrophobic surfaces as a passive-absorbing drag-reduction method. The turbulent flow is further investigated through flow visualizations and statistics of the relevant quantities, such as vorticity and strain rates. When rescaled in drag-reduction viscous units, the streamwise vortices over the hydrophobic surface are strongly altered, while the low-speed streaks maintain their characteristic spanwise spacing. We finally show that the reduction of vortex stretching and enstrophy production is primarily caused by the eigenvectors of the strain rate tensor orienting perpendicularly to the vorticity vector.

A firefly optimized fast extreme learning machine based maximum power point tracking for stability analysis of microgrid with two stage photovoltaic generation system

This paper proposes a new Firefly optimized Ridge Extreme Learning Machine (RELM-FF) algorithm based maximum power point tracking (MPPT) investigation for stability study of photovoltaic (PV) interactive microgrid dynamics. The proposed MPPT study is effective by managing minimum uncertainty limits, during solar uncertainty conditions (i.e., inconsistent irradiation, partial shading, etc.), compared with conventional techniques. A two stage converter dynamics is considered for the proposed grid interactive PV system. To withstand stability during worst grid operating scenario (both PV side and grid side uncertainty), a robust nonlinear Lyapunov based finite time sliding mode (LFSM) control is proposed for PV-VSC (voltage source converter) system. As the proposed multiple distributed energy resources based microgrid is considered as weak (low short circuit ratio), and PV is considered as an active power source, the Q-V based state variables are used for the dynamic control loop. A small signal stability analysis with the two stage PV converters is presented using the proposed RELM-FF algorithm and LFSM control. The performance evaluation of the microgrid operating with the PV array is demonstrated through various case scenarios, in MATLAB/Script environment. A hardware validation with TMS320C DSK board and embedded MATLAB coder is performed to establish effectiveness of the proposed MPPT algorithm, in real time. The superiority of the proposed MPPT over the conventional techniques is validated by comparing the stability improvement performance in grid interactive PV-VSC microgrid.

Estimation of the monthly precipitation predictability limit in China using the nonlinear local Lyapunov exponent

By using the nonlinear local Lyapunov exponent and nonlinear error growth dynamics, the predictability limit of monthly precipitation is quantitatively estimated based on daily observations collected from approximately 500 stations in China for the period 1960–2012. As daily precipitation data are not continuous in space and time, a transformation is first applied and a monthly standardized precipitation index (SPI) with Gaussian distribution is constructed. The monthly SPI predictability limit (MSPL) is quantitatively calculated for SPI dry, wet, and neutral phases. The results show that the annual mean MSPL varies regionally for both wet and dry phases: the MSPL in the wet (dry) phase is relatively higher (lower) in southern China than in other regions. Further, the pattern of the MSPL for the wet phase is almost opposite to that for the dry phase in both autumn and winter. The MSPL in the dry phase is higher in winter and lower in spring and autumn in southern China, while the MSPL values in the wet phase are higher in summer and winter than those in spring and autumn in southern China. The spatial distribution of the MSPL resembles that of the prediction skill of monthly precipitation from a dynamic extended-range forecast system.

Global robust adaptive path-tracking control of underactuated ships under stochastic disturbances

This paper presents a design of new controllers that force underactuated ships under both deterministic and stochastic sea loads to globally track a reference path. First, the loads are decomposed to a deterministic part treated as unknown constants, and a time-varying part considered as stochastic disturbances with unknown time-varying covariances. Second, the path-tracking errors are represented in a moving frame attached to the path. These errors are then to be stabilized at the origin by a design of controllers based on backstepping and Lyapunov׳s direct methods. Weak and strong nonlinear Lyapunov functions are introduced, and the path-parameter is used as an additional control input to overcome difficulties caused by underactuation and Hessian terms induced by stochastic differentiation rule. Estimate of the deterministic disturbances and upper covariances of the stochastic disturbances is incorporated in the control design. The effectiveness of the proposed results is illustrated through simulations.

On the Ψ-Conditional Asymptotic Stability of Nonlinear Lyapunov Matrix Differential Equations

Abstract It is proved (necessary and) sufficient conditions for Ψ − conditional asymptotic stability of the trivial solution of linear or nonlinear Lyapunov matrix differential equations.

Nonlinear Lyapunov criteria for stochastic explosive solutions

This paper deals with the problem of explosive solutions for a class of stochastic differential equations via Lyapunov’s method. We give some sufficient conditions on the existence of explosive solutions for stochastic differential equations driven by Brownian motions.

A New Generalized-Type of Synchronization for Discrete-Time Chaotic Dynamical Systems

In this paper, a new type of chaos synchronization in discrete-time is proposed by combining matrix projective synchronization (MPS) and generalized synchronization (GS). This new chaos synchronization type allows us to study synchronization between different dimensional discrete-time chaotic systems in different dimensions. Based on nonlinear controllers and Lyapunov stability theory, effective control schemes are introduced and new synchronization criterions are derived. Numerical simulations are used to validate the theoretical results and to verify the effectiveness of the proposed schemes.

Thermosolutal convection in a rotating couple-stress uid

In this paper we study the nonlinear Lyapunov stability of the conduction-diusion solution of a rotating couple-stress uid, in a layer heated and salted from below. After reformulating the perturbation evolution equations in a suit- able equivalent form, we derive the appropriate Lyapunov function and we prove that, if the principle of exchange of stabilities holds, the linear and nonlinear stability bounds are equal.The nonlinear stability bound is exactly the critical Rayleigh num-ber obtained solving the linear instability problem of the conduction- diusion solution.

Path-Tracking Control of Stochastic Quadrotor Aircraft in Three-Dimensional Space

Despite the fact that environmental loads (forces and moments) induced by wind on quadrotor vertical take-off and landing (VTOL) aircraft consist of both deterministic and stochastic components, all existing works on controlling the aircraft either ignore these loads or treat them as deterministic. This ignorance or treatment deteriorates the control performance in a practical implementation. This paper presents a constructive design of controllers for a quadrotor aircraft to track a reference path in three-dimensional (3D) space under both deterministic and stochastic disturbances. A combination of Euler angles and unit-quaternion for the attitude representation of the aircraft is used to result in an effective control design, and to achieve path-tracking control results. Weak and strong nonlinear Lyapunov functions are introduced to overcome difficulties caused by underactuation and Hessian terms induced by stochastic differentiation rule. To overcome the inherent underactuation of the aircraft, the roll and pitch angles of the aircraft are considered as immediate controls. Potential projection functions are introduced to design estimates of the deterministic components and covariances of the stochastic components. Simulations illustrate the results.

Nonlinear multiscale Maximal Lyapunov Exponent for accurate myoelectric signal classification

Surface Electromyography (sEMG) is a non-invasive, easy to record signal of superficial muscles from the skin surface. The sEMG is widely used in evaluating the functional status of the hand to assist in hand gesture recognition, prosthetics and rehabilitation applications. Considering the nonlinear and non-stationary characteristics of sEMG, hand gesture recognition using sEMG signals necessitate designers to use Maximal Lyapunov Exponent (MLE) or ensemble Empirical Mode Decomposition (EMD) based MLEs. In this research, we propose a hand gesture recognition method of sEMG based on nonlinear multiscale MLE. The aim is to increase the classification accuracy of sEMG features while reducing the complexity of EMD. The nonlinear MLE features are classified using Flexible Neural Tree (FNT), which can solve highly structured dependent problems of the Artificial Neural Network (ANN). The testing has been conducted using several experiments with five participants. The classification performance of nonlinear multiscale MLE method is compared with MLE and EMD-based MLE through simulations. Experimental results demonstrate that the former algorithm outperforms the two latter algorithms and can classify six different hand gestures up to 97.6% accuracy.

Global Path-Following Control of Stochastic Underactuated Ships: A Level Curve Approach

A level curve approach is introduced to design global path-following controllers for an underactuated surface ship. The approach is based on the observation: if the position of the ship satisfies the equation of the reference path, then the ship will be on the path. Thus, the controllers are designed based on Lyapunov's direct and backstepping methods to force the position of the ship to satisfy the equation of the path and to move along the path tangentially. The approach does not require computation of the position from the ship to the path. Weak and strong nonlinear Lyapunov functions are introduced in the control design to overcome difficulties caused by underactuation and to guarantee boundedness of the sway velocity. Simulations are included to illustrate the effectiveness of the proposed results.

On the Ψ-Strong Stability of Nonlinear Lyapunov Matrix Differential Equations

Abstract The paper provides (necessary and) sufficient conditions for Ψ-strong stability of the trivial solution of a linear Lyapunov matrix differential equations. Further, sufficient condition are obtained for Ψ-strong stability of the trivial solution of a nonlinear Lyapunov matrix differential equation.

Estimating the limit of decadal-scale climate predictability using observational data

Current coupled atmosphere–ocean general circulation models can not simulate decadal variability well, and model errors would have a significant impact on the estimation of decadal predictability. In this study, the nonlinear local Lyapunov exponent method is adopted to estimate the limit of decadal predictability based on 9-year low-pass filtered sea surface temperature (SST) and sea level pressure (SLP) observations. The results show that the limit of decadal predictability of the SST field is relatively large in the North Atlantic, North Pacific, Southern Ocean, tropical Indian Ocean, and western North Pacific, exceeding 7 years at most locations in these regions. In contrast, the limit of the SST field is relatively small in the tropical central–eastern Pacific (4–6 years). Similar to the SST field, the SLP field has a relatively large limit of decadal predictability over the Antarctic, North Pacific, and tropical Indian Ocean (>6 years). In addition, a relatively large limit of decadal predictability of the SLP field also occurs over the land regions of Africa, India, and South America. Distributions of the limit of decadal predictability of both the SST and SLP fields are almost consistent with those of their intensity and persistence on decadal timescales. By examining the limit of decadal predictability of several major climate modes, we found that the limit of decadal predictability of the Pacific decadal oscillation (PDO) is about 9 years, slightly lower than that of the Atlantic multidecadal oscillation (AMO) (about 11 years). In contrast, the northern and southern annular modes have limits of decadal predictability of about 4 and 9 years, respectively. However, the above limits estimated using time-filtered data may overestimate the predictability of decadal variability due to the use of time filtering. Filtered noise with the same spectral characteristics as the PDO and AMO, has a predictability of about 3 years. Future work is required with a longer period of observations or using a more realistic model of decadal variability to assess the real-time decadal predictability.

Stability analysis and fail-safe operation of inverters operated in parallel

In this paper, the stability in the sense of boundedness of inverters operated in parallel is proven and the fail-safe operation is achieved for generic linear or nonlinear loads described in the generalised dissipative Hamiltonian form. The robust droop controller (RDC), recently proposed in the literature for achieving accurate proportional power sharing and tight voltage regulation, is implemented in a nonlinear matrix form to achieve a bounded inverter voltage and inherit a fail-safe capability. Using nonlinear Lyapunov methods, it is proven that the RDC with fail-safe approximates the original RDC, generates a bounded inverter output voltage within the given technical limits and guarantees nonlinear closed-loop system stability in the sense of boundedness. When the prescribed limits are violated, e.g. due to sensor failure, the proposed method rapidly shuts down in a continuous-time manner, thus disconnecting the inverter to prevent a complete system failure even when the protection circuit fails. Extensive simulation results are presented to demonstrate this approach for two single-phase paralleled inverters feeding a linear and a nonlinear load under a sensor failure scenario.

Global path-following control of underactuated ships under deterministic and stochastic sea loads

SUMMARYThis paper presents a new method to design global path-following controllers for underactuated ships under both deterministic and stochastic sea loads. The path-following errors are first interpreted in a moving frame attached to the path. These errors are then to be stabilized at the origin by a design of controllers based on backstepping and Lyapunov's direct methods. Weak and strong nonlinear Lyapunov functions are introduced to overcome difficulties caused by underactuation and Hessian terms induced by stochastic differentiation rule, and to guarantee boundedness of the sway velocity. Potential projection functions are introduced to design update laws that provide bounded estimates of the mean values and covariances of the disturbances. Simulations are included to illustrate the effectiveness of the proposed approach.

Individual identification based on chaotic electrocardiogram signals during muscular exercise

An electrocardiogram (ECG) records changes in the electric potential of cardiac cells using a noninvasive method. Previous studies have shown that each person's cardiac signal possesses unique characteristics. Thus, researchers have attempted to use ECG signals for personal identification. However, most studies verify results using ECG signals taken from databases which are obtained from subjects under the condition of rest. Therefore, the extraction and analysis of a subject's ECG typically occurs in the resting state. This study presents experiments that involve recording ECG information after the heart rate of the subjects was increased through exercise. This study adopts the root mean square value, nonlinear Lyapunov exponent, and correlation dimension to analyse ECG data, and uses a support vector machine (SVM) to classify and identify the best combination and the most appropriate kernel function of a SVM. Results show that the successful recognition rate exceeds 80% when using the nonlinear SVM with a polynomial kernel function. This study confirms the existence of unique ECG features in each person. Even in the condition of exercise, chaotic theory can be used to extract specific biological characteristics, confirming the feasibility of using ECG signals for biometric verification.

New invariant measures to track slow parameter drifts in fast dynamical systems

Estimates of quantitative characteristics of nonlinear dynamics, e.g., correlation dimension or Lyapunov exponents, require long time series and are sensitive to noise. Other measures (e.g., phase space warping or sensitivity vector fields) are relatively difficult to implement and computationally intensive. In this paper, we propose a new class of features based on Birkhoff ergodic theorem, which are fast and easy to calculate. They are robust to noise and do not require large data or computational resources. Application of these metrics in conjunction with the smooth orthogonal decomposition to identify/track slowly changing parameters in nonlinear dynamical systems is demonstrated using both synthetic and experimental data.