Statistical Linearization Technique Research Articles

Towards efficient control synthesis for nonlinear wave energy conversion systems: impedance-matching meets the spectral-domain

Existing studies within the literature that focus on designing parametric energy-maximizing controllers for Wave Energy Converter (WEC) systems predominantly rely on the impedance-matching (IM) principle, originally developed for linear time-invariant systems. Alternatively, iterative optimization routines are commonly employed for nonlinear WECs. However, these approaches often face a trade-off between effectiveness in maximizing energy extraction and computational efficiency. To address this limitation, this study proposes a computationally efficient controller tuning method for analogous synthesis in the case of nonlinear WECs. The proposed approach combines a statistical linearization technique known as spectral-domain modeling with the IM principle, to synthesize a Proportional–Integrative (PI) controller for a nonlinear WEC. Furthermore, a comparison is performed with two other synthesis methods: one based on a standard (i.e. linear) frequency-domain representation of the WEC that incorporates the IM principle, and the other employing a gradient-free optimization routine applied to the nonlinear time-domain model of the WEC for PI parameter tuning through exhaustive numerical search. A discussion on the effectiveness of each tuning method in maximizing energy absorption is provided, including an appraisal of their associated computational time requirements. Numerical analyses demonstrate that the proposed method, which integrates spectral-domain modeling and IM, can achieve (almost) optimal PI controller design for a nonlinear WEC. Furthermore, this study addresses the inaccuracies inherent in the frequency-domain approach and significantly reduces the computational time compared to the exhaustive search procedure. The findings of this research represent a significant advancement towards the development of simple, effective, and efficient IM-based techniques for synthesis of controllers in nonlinear WEC systems

Sequential track fusion in multi-sensor networks of unscented Kalman filters: A case of slip estimation in planetary mobile robots

Accurate wheel slip estimation facilitates Wheeled Mobile Robots (WMRs) with improved localization and traversability monitoring which are crucial to their autonomy in challenging planetary environments. Although distributed track-level fusion offers better computational efficiency, often sensor-level fusion is adopted in slip estimators of planetary rovers. Extending upon our earlier work, we develop a novel explicit track-to-track fusion algorithm for multi-sensor networks of unscented Kalman filters that has immediate application to slip ratio estimation in WMRs. The algorithm demonstrates better consistency compared to the existing fusion techniques since it implements a sub-optimal fusion rule to sequentially combine local tracks. It also saves computational power due to the recursive propagation of cross-covariance matrices based on the statistical linearization technique, instead of performing online optimizations. We prove that this recursion only requires the information of the first and last tracks in a sequence if all local tracks are unbiased. We rigorously study various key properties of the developed fusion algorithm and show its superior level of confidence when compared to the two prominent fusion methods of sequential and batch covariance intersection. The slip ratio estimation in a six-wheel WMR is considered as a case study, where the steerable wheel sets act as a network of sensors. The proposed slip estimator works based on the rigid body kinematics and readings of purely proprioceptive sensors, i.e., an inertial measurement unit and encoders. The slip estimator’s performance is evaluated in a high-fidelity software-in-the-loop simulation environment connecting MATLAB and CM Lab’s Vortex Studio software. In a comparison study that considers four rival strategies, we show the superiority of the proposed fusion method offering a balance between consistency, accuracy, and speed in real-time slip estimations.

Optimal design of tuned inerter negative stiffness dampers with linear viscous and nonlinear eddy current damping for structural effective damping ratio enhancement

Tuned inerter negative stiffness damper (TINSD) has demonstrated its effectiveness in controlling structural vibrations. However, the potential to improve the effective damping ratio of primary structures remains unexplored, and thus far, TINSD has solely featured linear viscous damping. In this study, we explored two damping types for TINSD: linear viscous damping (abbreviated as TINSD) and nonlinear eddy current damping (termed EC-TINSD). The optimal design of the TINSD was conducted based on the effective damping ratio enhancement (EDRE) effect. Our findings reveal that an increase in the absolute value of the negative stiffness does not result in a continuous improvement in the control efficiency and effectiveness of TINSD, which correspond to the EDRE effect and the structural effective damping ratio, respectively. Importantly, we observe that the proposed optimal TINSD solution outperforms the one derived from the fixed-point method, particularly when the inertance ratio is less than 0.15. For EC-TINSD, its optimal design was performed based on the EDRE effect using statistical linearization techniques that employ both force-based and energy-based equivalent criteria. The control efficiency and effectiveness of optimal EC-TINSD were then validated through Monte Carlo simulation. This analysis highlights the significance of a large critical velocity for EC-TINSD to achieve greater accuracy in vibration control. Numerical studies were further conducted to investigate the control efficiency and effectiveness of EC-TINSD under three types of real seismic excitations. The results demonstrate that both TINSD and EC-TINSD exhibit similar effectiveness in mitigating the maximum seismic responses of primary structures, and their efficiency surpasses that of viscous dampers and eddy current dampers with identical damping parameters. Additionally, the distinctive damping force limiting characteristic of EC-TINSD should be commended as it can protect the security and dependability of both the dampers and their connections to the structures.

Nonstationary Stochastic Response of Hysteretic Systems Endowed With Fractional Derivative Elements

AbstractIn this paper, a computationally efficient approach is proposed for the determination of the nonstationary response statistics of hysteretic oscillators endowed with fractional derivative elements. This problem is of particular practical significance since many important engineering systems exhibit hysteretic/inelastic behavior optimally captured only through the concept of fractional derivative, and many natural excitations as seismic waves and atmospheric turbulence are both stochastic and nonstationary in time. Specifically, the approach is based on a statistical linearization scheme involving an equivalent system of augmented dimension. First, relying on a transformation scheme, the fractional derivative term is represented by a set of coupled linear ordinary differential equations. Next, the evolution of the system response statistics is captured by incorporating the statistical linearization technique in a nonstationary sense. This involves integrating in time a set of ordinary differential equations. Several numerical applications pertaining to classical hysteretic oscillators are considered, and the versatility of the proposed method is assessed via comparison with pertinent Monte Carlo simulations.

Statistical linearization for random vibration energy harvesting with piezoelectric material nonlinearity

Due to piezoelectric softening and dissipative nonlinearities, the piezoelectric cantilever energy harvester exhibits nonlinear hysteresis when subjected to large excitation. These nonlinearities have brought significant challenges to the modeling and response prediction of randomly excited piezoelectric energy harvesting systems. In this study, the voltage responses of the nonlinear piezoelectric cantilever energy harvester under random excitation are initially assumed to follow the Gaussian distribution which is experimentally validated later. The equivalent linear transfer function are derived from the approximate linearization of the stiffness and damping using the statistical linearization (SL) technique. The mathematical expectations of the voltage responses are calculated from the multivariate normal distributions. Frequency sweep experiments are conducted to a cantilever energy harvester to identify the nonlinear piezoelectric material properties. The statistically linearized model was experimentally validated under random base acceleration excitation by comparing the probability density function of the predicted voltage responses and average power against the experimental measurements. The advantage of the SL technique lies in allowing one to use an iterative procedure to estimate the equivalent linear terms while the analytical expressions are unattainable because of the complex nonlinearity in the governing equations. The results show that the prediction of the SL model to the random base acceleration excitation agrees with experimental measurements with a broadband frequency range, although only the fundamental mode of the beam is considered.

Application scope of equivalent linear technique for across-wind response analysis of tall flexible structures considering nonlinear aero-elastic effect

Large across-wind response of tall flexible structures becomes an increasing concern due to the significant nonlinear aero-elastic effect. The nonlinear aerodynamic damping, which is in phase of time-variant structural velocity, is usually employed to describe nonlinear aero-elastic effect. For the sake of evaluating the across-wind response of tall flexible structures within the framework of linear random vibration theory, equivalent linear aerodynamic damping model is required, which can be deduced from either harmonic balance (HB)- or statistical linearization (SL) technique. This study addressed clarifying application scope of various equivalent linear techniques for evaluating across-wind response of tall flexible structures. By conducting the Monte Carlo simulation and free-vibration wind tunnel test, it is clarified that the root-mean-squared (RMS) response calculated by the HB equivalent linear aerodynamic damping is acceptable, while is not satisfactory with respect to the SL technique especially when the structure is with lower damping level, and in the vicinity of vortex lock-in wind speed. Moreover, it is found that neither HB- nor SL technique can accurately predict the peak response. It is revealed in this study that accuracy of various equivalent linear techniques is essentially affected by probability density function of across-wind response.

Develop and Verify Energy-Based Statistical Linearization Technique to Analysis Nonlinear Stochastic Vibration of a Spur Gear Pair

Abstract In this article, an energy-based statistical linearization method (SL) is proposed to simulate a nonlinear dynamic model of spur gear pair. The gear pair operates under combined deterministic and random loads, and both backlash and time-varying mesh stiffness are considered in the dynamic model. The equivalent linear function approximates the teeth backlash nonlinearity in the gear model. The energy-based linearization, which minimizes the error in potential energy between the original and equivalent linear systems, is used. Simulations are conducted on a gear pair, and the effect of the input torque on the dynamic response of the gear pair is then examined. The results demonstrate that for high input torque, the system operates in the linear range. For low input torque, the results are not similar to the original because the system became strongly nonlinear. Monte Carlo simulations were carried out to verify the accuracy of the presented method.

Nonstationary stochastic response determination of nonlinear oscillators endowed with fractional derivatives

Fractional calculus has been broadly used in diverse engineering applications. In this regard, vibrations of fractional oscillators subject to stochastic loads have attracted considerable attention. This paper proposes a semi-analytical approach to determine the nonstationary response statistics of nonlinear oscillators endowed with fractional derivatives. Specifically, the fractional derivative term is represented by introducing a discretization scheme and associated first-order differential equations. This leads to an augmented dimension dynamic system. Further, in conjunction with the statistical linearization technique, the evolution of the system response is captured by a set of coupled ordinary differential equations with time-dependent coefficients. Furthermore, solving the associated Lyapunov equation for the randomly excited dynamic system yields the nonstationary statistics of the oscillator response. The reliability of the proposed method is demonstrated by Monte Carlo simulations pertaining to classical nonlinear oscillators.

Stochastic response determination of U-OWC energy harvesters: a statistical linearization solution treatment accounting for intermittent wave excitation

A novel statistical linearization technique is developed for determining approximately the response statistics and the power output of U-Oscillating Water Column (U-OWC) energy harvesting systems. In this regard, first, the governing equations are derived by employing the unsteady Bernoulli equation. Note that the intermittent, i.e., non-stationary, nature of the wave excitation, occurring in severe sea states due to uncovering of the U-OWC inlet, is explicitly accounted for in the herein proposed model. This is done by multiplying the excitation process with a Heaviside function dependent on the instantaneous free surface displacement. Next, the resulting coupled system of nonlinear integro-differential stochastic equations is solved approximately by relying on a statistical linearization technique. Specifically, the original system of nonlinear equations is replaced by an equivalent linear one, whose parameters and response first- and second-order statistics are obtained by minimizing the mean square error between the two systems. A significant novel aspect of the technique relates to the fact that the Heaviside function is replaced in the equivalent linear system by an “equivalent excitation” coefficient to be determined as part of the statistical linearization solution scheme. Further, compared with other relevant solution schemes adopted in earlier research efforts in the literature, it is shown that the developed technique can be construed as a direct generalization that exhibits an enhanced accuracy degree. The U-OWC installed in the Civitavecchia harbor (Rome, Italy) is considered as an illustrative numerical example, where the reliability of the approximate technique is demonstrated by comparisons with pertinent Monte Carlo simulation data.

Robust stable iterated unscented Kalman filter based on maximum correntropy criterion

The Unscented Kalman filter (UKF) based on maximum correntropy criterion (MCC) is robust to heavy-tailed non-Gaussian noise. However, the approximate linear measurement equation obtained by statistical linearization technique may be not accurate enough since it only uses prior information. In this paper, a robust stable iterative maximum correntropy criterion UKF (RS-IMCC-UKF) is proposed by using nonlinear measurement function directly and numerical stability methods. Different from the existing UKF algorithms, we only need to perform one-time unscented transformation in each filtering cycle, reducing the execution time of algorithm. Then a nonlinear enhancement model is constructed to handle predictions and observations simultaneously, which will be included in the cost function of robust MCC. In the process of iterative solution, thanks to the latest iteration values which are used to update the measurement information, RS-IMCC-UKF has more accurate results compared with traditional filters. In order to avoid non-positive definite characteristic in covariance matrix and the inverse operation in ill-conditioned matrix, the hyperbolic QR decomposition and Moore–Penrose pseudo-inversion are introduced to improve the numerical stability of the algorithm. Finally, a target tracking is modeled to verify the effectiveness of the algorithm.

Performance enhancement of FPS-isolated buildings using an inerter-based damper: Stochastic seismic analysis and optimization

Enhancing the seismic performance of based-isolated structures during a major earthquake event using inerter-based dampers emerges as a prominent research direction. However, analysis and optimization of based-isolated structures incorporated with inerter-based dampers considering both system nonlinearity and excitation uncertainty have rarely been reported. An electromagnetic inertial mass damper (EIMD) is a high-performance passive damper that consists of an inerter element and a damping element in parallel, and its superior control performance has been validated when applied to stay cables and frame structures. This paper presents a stochastic seismic analysis and parameter optimization of a friction pendulum system (FPS)-isolated structure with an EIMD considering the Kanai-Tajimi earthquake model. Considering the nonlinear damping of the FPS system, we derive the semi-analytical solutions of the FPS-isolated structure with an EIMD under stationary stochastic excitations based on a two-degree-of-freedom model using the statistical linearization technique. The effect of the EIMD on the FPS-isolated structure is systematically investigated via parametric study. A numerical procedure is proposed for optimizing the parameters of the EIMD and the FPS for minimizing the seismic response variances. Numerical simulation of a seven-story FPS-isolated building subjected to both artificial and real seismic excitations illustrates the effectiveness of the optimization procedure based on the semi-analytical solutions. Numerical results also reveal that the EIMD is capable of reducing both the base floor responses and superstructure responses, which outperforms tuned inerter dampers and conventional fluid viscous dampers. This study might shed light on the applications of inerter-based dampers in based-isolated structures in the years to come.

Development of novel track nonlinear energy sinks for seismic performance improvement of offshore wind turbine towers

High-rise and flexible offshore wind turbines (OWTs) are frequently constructed in high-seismicity regions, and passive linear control strategies are commonly adopted to mitigate the seismic responses of OWTs. However, the effectiveness of linear control devices is often sensitive to the frequency contents of seismic motions as well as structural frequencies. Track nonlinear energy sinks (NESs) have recently been proven effective in broadband vibration suppression but also possess amplitude-dependent control performance. Two novel NESs with newly-designed track profiles combining quadratic and quartic polynomials are proposed in the present study to improve the effectiveness of track NESs. One (hybrid track NES) combines nonlinear and linear positive stiffness, whereas the other (bistable track NES) combines nonlinear positive and linear negative stiffness. A statistical linearization technique is adopted to optimize the profiles and damping coefficients of the track NESs. A 3D finite element model of a typical OWT is established, and the restoring force composition of the track NESs is analyzed to simplify their numerical modeling. The effectiveness and robustness of the hybrid and bistable track NESs in OWT vibration control are systematically investigated under a set of synthetic and real ground motions considering different structural frequencies and further compared to the tuned mass damper (TMD) and the traditional track NES with a fourth-order polynomial profile. Numerical results show that hybrid and bistable track NESs outperform their counterpart of the traditional track NES. NESs exhibit a robust behavior against frequency changes despite their slightly less control effectiveness than that of the TMD. Moreover, large response reductions are achieved using the bistable track NES under low and moderate input energy levels.

Robust Power System Forecasting-Aided State Estimation With Generalized Maximum Mixture Correntropy Unscented Kalman Filter

As an effective method for power system forecasting-aided state estimation (FASE), the unscented Kalman filter (UKF) based on correntropy has been widely used in recent years, ensuring the safe and reliable operation of power systems. In this article, to address the impulsive noise, Laplacian noise, bad measurement data, and sudden load change, a robust UKF algorithm based on generalized maximum mixture correntropy (GMMC-UKF) criterion is proposed for FASE, in which the kernel is composed of two generalized Gaussian functions. Specifically, we use a statistical linearization technique to unify the state error and measurement error in the cost function and obtain the optimal value of state estimation by fixed-point iteration. The effectiveness of the proposed algorithm for FASE is verified on IEEE 14-, 30-, and 57-bus test systems under a variety of abnormal situations. Compared with traditional correntropy algorithms, the GMMC-UKF shows more accurate estimation and stronger robustness.

Optimal design of tuned liquid column damper inerter for vibration control

In this paper, the use of a novel passive control device defined as Tuned Liquid Column Damper Inerter (TLCDI) is studied to control the seismic response of structural systems. The TLCDI, recently introduced as an enhanced version of the conventional Tuned Liquid Column Damper, may achieve improved seismic performances by exploiting the mass amplification effect of the so-called inerter device. For this purpose, an optimization procedure for the design of the TLCDI based on a statistical linearization technique and the minimization of the structural displacement variance is proposed. Notably, by assuming a white noise base excitation and considering some additional approximations, pertinent closed-form expressions for the optimal TLCDI parameters are provided. The reliability of the proposed analytical solutions is proved by a comparison with numerical results obtained by a more computationally demanding iterative optimization technique on the original damped system. Finally, the efficiency of the control performance of TLCDI-controlled structures is examined using real recorded seismic signals as external excitation.

Nonlinear stochastic vibration of a variable cross-section rod with a fractional derivative element

The article deals with the problem of computing efficiently the nonlinear response of a rod involving a fractional constitutive model, and exposed to random excitation. The constitutive model is a three-parameter model comprising an instant elasticity modulus, a prolonged elasticity modulus, and a relaxation parameter. The nonlinear term is a linear-plus-cubic force of the Winkler kind. The resulting nonlinear fractional partial differential equation governing the rod displacement has no known exact solution. Thus, the article proposes an approximate analytical solution by relying on the statistical linearization technique. Further, it develops a Boundary Element Method (BEM)-based approach to estimate numerically the rod response statistics. The statistical linearization solution is obtained by representing the rod displacement as the superposition of linear modes of vibration having time-dependent coefficients. In this context, it is shown that the equation governing the time variation of the mode coefficients is a nonlinear fractional ordinary differential equation, whose solution is computed by a surrogate linear system identified by minimizing the response error between the linear system and the nonlinear one in a mean square sense. Relevant Monte Carlo studies pertaining to rods with fixed-fixed, and fixed-free ends show that the proposed analytical solution is in a good agreement with data obtained by the numerical (BEM) approach.

A prey-predator model with Holling type IV response function under deterministic and stochastic environment

In this paper, both deterministic and stochastic behaviors of a general prey-predator model have been studied with Holling type-IV response function. For the deterministic model, uniform boundedness and persistence of the system have been discussed under the certain condition of the parameter. For local stability and bifurcation analysis, we arrive at the Hopf bifurcation and derived the symbolic condition for Hopf bifurcation. After that, the model has been illustrated with some numerical examples. In the second phase, the system has been perturbed by independent Gaussian white noises for the stochastic environment and the stability of the system have been studied by statistical linearization technique. Finally, a comparison has been made between the stability conditions in deterministic and stochastic cases.

Harmonic wavelets based response evolutionary power spectrum determination of linear and nonlinear structural systems with singular matrices

A new approximate analytical technique is proposed for determining the response evolutionary power spectrum (EPS) of stochastically excited structural multi-degree-of-freedom (MDOF) linear and nonlinear systems with singular matrices. Such systems can appear, indicatively, when a redundant coordinates modeling is adopted for forming the equations of motion of complex multi-body systems. For this case, it can be argued that this modeling approach facilitates the system’s stochastic response analysis, since employment of redundant DOFs is associated with computational cost efficient solution frameworks, and potentially provides with enhanced modeling flexibility. In this context, aiming at the joint time–frequency response analysis of MDOF systems, recently developed wavelet-based solution frameworks, which generalize classic input–output relationships of random vibration, are adopted and further generalized in this paper to account for systems with singular matrices. Specifically, resorting to the theory of generalized inverses of singular matrices, as well as to the theory of harmonic wavelets, a Moore–Penrose generalized matrix inverse excitation-response relationship is derived herein for determining the response EPS of linear MDOF systems. Further, a recently developed harmonic-wavelet-based statistical linearization technique is also generalized to account for the case of nonlinear MDOF systems. The validity of the proposed technique is demonstrated by pertinent numerical examples.

Development and verification of a computationally efficient stochastically linearized planetary gear train model with ring elasticity

This paper develops a hybrid dynamic model for the analysis of nonlinear stochastic vibration of a wind turbine’s planetary gear train (PGT) with an elastic ring gear. The elastic deformation of ring gear is considered and incorporated into the model through the finite element model. Moving loads are used to incorporate the meshing loads between the planet-ring gear pairs. In addition, the model includes both the stiffness variation and the backlash nonlinearity between meshing gears. The backlash is linearized using the statistical linearization technique, and a linear set of equations is obtained based on which simulation is carried out with PGT parameters in wind turbines. A parametric study is conducted, and the results demonstrate that the statistical linearization method is accurate enough to study the nonlinear stochastic PGT. The results reveal that the moving mesh has a strong effect on PGT when the ring gear elasticity is considered. The influence of rim thickness on PGT dynamics was analyzed, and the results show that the effect is significant. Monte Carlo simulations are used to verify the accuracy of the proposed method.

Statistical linearization of the Morison’s equation applied to wave energy converters

The viscous drag acting on wave energy converters may have a significant effect on the dynamics during high-energetic sea states and large motions experienced due to resonance. The viscous drag is a nonlinear phenomenon of floating systems usually modelled based on the Morison’s equation using the relative velocity between the structure and the wave particle. To include such a nonlinearity into the system dynamics, nonlinear time domain simulations are generally employed, which are computationally expensive compared to frequency domain simulations. To overcome this problem, this work presents the derivation of the viscous drag force/torque under the statistical linearization technique using the frequency domain model. The technique offers a reliable tool for the estimation of the system dynamics while maintaining a low computational cost when compared to time domain simulations. For the proposed nonlinearity, the resulting equivalent linear term can be decomposed into two components: an excitation term and a damping term. To illustrate the applicability of the derivation, two conceptually different wave energy converters are investigated: a heaving point absorber, and an oscillating wave surge converter. The results obtained using statistical linearization are compared to their respective nonlinear time domain simulations to verify the reliability of the technique. Also, a comparison between the statistical linearization results using the relative motion and using only the structure motion is presented to illustrate the importance of including the relative velocity for wave energy applications. Excellent agreements have been obtained between statistical linearization model using the relative motion and its respective nonlinear time domain model for both devices in terms of spectral content, probability density of the velocity components, and energy absorbed by the device.

Efficient computation of optimal open-loop controls for stochastic systems

Optimal control is a prominent approach in robotics and movement neuroscience, among other fields of science. Methods for deriving optimal choices of action have been classically devised either in deterministic or stochastic settings. Here, we consider a setting in-between that retains the stochastic aspect of the controlled system but assumes deterministic open-loop control actions. The rationale stems from observations about the neural control of movement which highlighted that relatively stable behaviors can be achieved without feedback circuitry, via open-loop motor commands adequately tuning the mechanical impedance of the neuromusculoskeletal system. Yet, effective methods for deriving optimal open-loop controls for stochastic systems are lacking overall. This work presents a continuous-time approach based on statistical linearization techniques for the efficient computation of optimal open-loop controls for a broad class of stochastic optimal control problems. We first show that non-trivial departure from the optimal solutions of classical deterministic and stochastic approaches may arise for simple synthetic examples, thereby stressing the originality of the framework. We then exemplify its potential relevance to the planning of biological movement by showing that a well-known phenomenon in motor control, referred to as muscle co-contraction, occurs naturally. More generally, this stochastic optimal control framework may be suited to other fields where the design of optimal open-loop actions is relevant.