Saturation-type Nonlinearity Research Articles

Displacement tracking of uncertain nonlinear cardiovascular muscle using Lyapunov function with disturbance observer-based control

The nonlinear behaviour of the human cardiovascular muscle is examined in this work through simulation. Non-zero reaction time of the proposed design includes the dead zone nonlinearity while saturation type nonlinearity comes after some linear region. When a force is applied to the cardiovascular muscle, an uncertain displacement is observed due to the nonlinear nature which is analysed using robust control theory. The stability is investigated using 'Lyapunov stability theory' along with observer based control system. A nonlinear controller has also been designed in the absence of uncertainty and any other disturbances to track the displacement of muscle dynamics. To achieve desired tracking performances, stability analysis has successfully been done using appropriate simulation that ensures the reliability of the proposed nonlinear model.

Statistical-Mechanical Analysis of Adaptive Filter With Clipping Saturation-Type Nonlinearity

In most practical adaptive signal processing systems, e.g., active noise control, active vibration control, and acoustic echo cancellation, substantial nonlinearities that cannot be neglected exist. In this paper, we analyze the behaviors of an adaptive system in which the output of the adaptive filter has the clipping saturation-type nonlinearity by a statistical-mechanical method. We discuss the dynamical and steady-state behaviors of the adaptive system by asymptotic analysis, steady-state analysis, and numerical calculation. As a result, it has become clear that the saturation value has the critical point at which the system's mean-square stability and instability switch. The obtained theory well explains the strange behaviors around the critical point observed in the computer simulation. Finally, the exact value of the critical point is also derived.

A Novel Approach for Improved Noise Reduction Performance in Feed-Forward Active Noise Control Systems With (Loudspeaker) Saturation Non-Linearity in the Secondary Path

In active noise control systems, the performance of the most celebrated filtered-x-least mean square (FxLMS) adaptive algorithm is degraded in the presence of non-linearity in the secondary path. In this article, we propose a novel approach to improve the performance of FxLMS algorithm in the presence of saturation type non-linearity in the secondary path. In the proposed method, the weights of the controller are adapted by minimizing a cost function consisting of two terms. The first term is the squared instantaneous residual error while the second term is the weighted square of the difference between input and output of (estimated) saturation non-linearity. Consequently, in case the saturation has occurred, the resultant adaptive algorithm minimizes the residual error while keeping the controller output close to its saturation limit. This avoids unnecessary fluctuation of the (controller) weights due to large variation (or windup) of the controller output after the saturation has occurred. Numerical simulations are provided to verify the superiority of the reported approach as compared to existing methods under different conditions.

Theoretical transient analysis of a hearing aid feedback canceller with a saturation type nonlinearity in the direct path

Feedback cancellation in a hearing aid is essential for achieving high maximum stable gain to compensate for the losses in severe to profound hearing impaired people. The performance of adaptive feedback cancellers has been studied by assuming that the feedback path can be modeled as a linear system. However, limited dynamic range, low-cost loudspeakers, and nonlinear power amplifiers may distort the hearing aid output signal. In this way, linear-based predictions of the canceller performance may lead to significant deviations from its actual behavior. This work presents a theoretical performance analysis of a Least Mean Square based shadow filter that is applied to set up the coefficients of a feedback canceller, which is subject to a static saturation type nonlinearity at the output of the direct path. Deterministic recursive equations are derived to predict the mean square feedback error and the mean coefficient vector evolution between updates of the feedback canceller. These models are defined as functions of the canceller parameters and input signal statistics. Comparisons with Monte Carlo simulations show the provided models are highly accurate under the considered assumptions. The developed models allow inferences about the potential impact of an overdriven loudspeaker over the transient performance of the direct method feedback canceller, serving as insightful tools for understanding the involved mechanisms.

Evidence for nonisochronism of material vibrations in stimulated Raman scattering

Evidence is found for a decrease of the Raman shift in stimulated Raman scattering with an increase of the pump intensity, which is interpreted as a manifestation of vibrations’ nonisochronism. It is shown that the nonisochronism corresponds to a saturation-type nonlinearity of material vibrations in fused silica and to a Kerr-like nonlinearity in BK-7 glass.

Open-Loop Control of Combustion Instabilities and the Role of the Flame Response to Two-Frequency Forcing

Controlling combustion instabilities by means of open-loop forcing at non-resonant frequencies is attractive because neither a dynamic sensor signal nor a signal processor is required. On the other hand, since the mechanism by which this type of control suppresses an unstable thermoacoustic mode is inherently nonlinear, a comprehensive explanation for success (or failure) of open-loop control has not been found. The present work contributes to the understanding of this process in that it interprets open-loop forcing at non-resonant frequencies in terms of the flame’s nonlinear response to a superposition of two approximately sinusoidal input signals. For a saturation-type nonlinearity, the fundamental gain at one frequency may be decreased by increasing the amplitude of a secondary frequency component in the input signal. This effect is first illustrated on the basis of an elementary model problem. In addition, an experimental investigation is conducted at an atmospheric combustor test-rig to corroborate the proposed explanation. Open-loop acoustic and fuel-flow forcing at various frequencies and amplitudes is applied at unstable operating conditions that exhibit high-amplitude limit-cycle oscillations. The effectiveness of specific forcing parameters in suppressing self-excited oscillations is correlated with flame response measurements that include a secondary forcing frequency. The results demonstrate that a reduction in the fundamental harmonic gain at the instability frequency through the additional forcing at a non-resonant frequency is one possible indicator of successful open-loop control. Since this mechanism is independent of the system acoustics, an assessment of favorable forcing parameters, which stabilize thermoacoustic oscillations, may be based solely on an investigation of burner and flame.

On Error-Saturation Nonlinearities in NLMS Adaptation

The effect of a saturation-type error nonlinearity in the weight update equation in normalized least mean-square (NLMS) adaptation is investigated for system identification for a white Gaussian data model. Nonlinear recursions are derived for the weight mean error and mean-square deviation (MSD) that include the effect of an error function (erf) saturation-type nonlinearity on the error sequence driving the algorithm. The nonlinear recursion for the MSD is solved numerically and shown in excellent agreement with Monte Carlo simulations, supporting the theoretical model assumptions. The theory is extended to tracking a Markov channel and accurately predicts the tracking behavior as well. The saturation behavior of the algorithm is easily studied by varying a single parameter in the error function, varying from a linear device to a hard limiter. For the white data case, the excess mean square-error (EMSE) is simply related to the MSD. The tradeoff between the extent of error saturation, steady-state EMSE, and algorithm convergence rate is studied using these results.

The Continuous System Equivalent to the System with Sliding Mode Control

It is shown how to create the continuous system equivalent to the system with sliding mode control. In the case of minimum phase plants, the system arises from the replacement of the relay by some appropriate saturation-type nonlinearity. The latter may be implemented using the amplifier with high gain and saturation with constraints of the control determined by the switched magnitudes of the relay. In the case of nonminimum phase plants it is noted that similar equivalence exists for the continuous and relay system with parallel compensator. The latter system may be treated as the system with modified sliding mode control. In the equivalent continuous system the chattering effect, related with sliding mode control doesn't exist.

Adaptive LQ Control With Anti-Windup Augmentation to Optimize UAV Performance in Autonomous Soaring Applications

The performance of a glider can be considerably improved by means of soaring and optimization of its trajectory. Recent advances in the area of static soaring assume known linear glider dynamics and no actuator saturation phenomena. In practice, the dynamics of the glider change with flight and environmental conditions, and the actuators moving the control surfaces have mechanical limits. In this paper, we consider the optimization-based static soaring problem in the presence of actuator saturation nonlinearities and large parametric uncertainties in the dynamics of the vehicle. We use ideas from robust adaptive control and anti-windup design tools to develop an adaptive control scheme based on linear quadratic control with disturbance rejection. The saturation-type nonlinearity is addressed by the proposed adaptive version of a linear matrix inequality based anti-windup design. The resulting adaptive control scheme with adaptive anti-windup augmentation allows optimal soaring despite the presence of significant actuator saturation limits and unknown parameters.

On the LMS algorithm with constant and variable leakage factor in a nonlinear environment

This paper studies the application of the linear least-mean-square (LMS) algorithm with leakage to a system having a memoryless saturation-type nonlinearity. This approach represents an interesting alternative to the nonlinear LMS (NLLMS) algorithm. The major drawback to implement the latter is that the model parameters of the system nonlinearity must be known. In contrast, the linear LMS algorithm with leakage does not require such knowledge. It permits to approximate the performance of the NLLMS by properly selecting a constant leakage factor value. To cope with an eventual change of the environment, a strategy for a variable leakage is also proposed. Several numerical simulations are presented with the aim of ratifying the feasibility of the proposed approach

Oscillation and convergence behaviours exhibited in an ‘unstable’ second‐order digital filter with saturation‐type non‐linearity

AbstractThis letter explains the oscillatory behaviours exhibited in a second‐order digital filter with saturation‐type non‐linearity via the Hopf bifurcation theorem. It is shown that depending on the bifurcation parameter, the state variables may converge to zero even when the eigenvalues of the system matrix are outside the unit circle. Copyright © 2004 John Wiley & Sons, Ltd. abstract

On a model for optical pulse propagation with saturation nonlinearity

We have suggested a new nonlinear equation for optical pulse propagation in the nonlinear medium with saturation type nonlinearity. The equation can be exactly analysed by Hirota's approach and we have studied the form of explicit one and two solition states.

IMD prediction techniques for active filters

Two single-domain techniques for evaluating the output nonlinear distortion of active systems with multitone excitations are described as a means of circumventing the time-frequency transformations of standard harmonic balance methods. These two techniques are compared with respect to accuracy, computational efficiency, and flexibility for a band-pass active filter containing a saturation-type nonlinearity.

Optical solitons in semiconductor-doped glass fibres with saturation-type nonlinearity

We study soliton solutions of the nonlinear equation governing pulse propagation in semiconductor-doped glass (SDG) fibres which have saturation-type nonlinearity. The phase modulation of the pulse is nonlinear and no sech-type soliton exists in SDG fibres when high-order correction terms are taken into account. As compared with optical solitons in ordinary glass fibres, the property of the solitons discussed here is somewhat different.

Resolving manipulator redundancy under inequality constraints

Due to hardware limitations, physical constraints such as joint rate bounds, joint angle limits, and joint torque constraints always exist. In this paper, these constraints are considered in the general formulation of the redundant inverse kinematic problem. To take these physical constraints into account, the computationally efficient compact quadratic programming (QP) method is formed to resolve the constrained kinematic redundancy problem. In addition, the compact-inverse QP method is also formulated to remedy the inescapable singularity problem with inequality constraints. Two examples are given to demonstrate the generality and superiority of these two methods: to eliminate the drift phenomenon caused by self motion and to remedy saturation-type nonlinearity problems. >

Effects of saturation on optical bistability with coupled surface plasmons

We present exact numerical results for a symmetric layered medium (prism/Ag film/nonlinear dielectric/Ag film/prism) where the middle dielectric slab is assumed to have a saturation-type nonlinearity. We show bistable behaviour in the power dependence of the reflectivity ofp-polarized light under the condition when coupled surface modes are excited in the structure. Moreover, we study the effect of saturation on the bistable behaviour to show that multivalued character is inhibited by saturation effects. The field distributions corresponding to the minimum reflectivity states of the nonlinear structure are also presented.

On weight update saturation nonlinearities in LMS adaptation

The effect of a saturation-type nonlinearity in the weight update equation in least mean square (LMS) adaptation is investigated for a white Gaussian data model. Nonlinear difference equations that include the effect of a 1-e/sup -ax/ saturation-type nonlinearity on the update term driving the algorithm are derived for the weight first and second moments. A nonlinear difference equation for the mean norm is explicitly solved using a differential equation approximation and integration by quadratures. The steady-state second-moment weight behavior is evaluated approximately for the nonlinearity. Using these results, the tradeoff between the extent of weight update saturation, steady-state excess mean square error, and rate of algorithm convergence is studied. For the same steady-state misadjustment error, the tradeoff results are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On error-saturation nonlinearities in LMS adaptation

The effect of a saturation-type error nonlinearity in the weight update equation in least-mean-squares (LMS) adaptation is investigated for a white Gaussian data model. Nonlinear difference equations are derived for the eight first and second moments, which include the effect of an error function (erf) saturation-type nonlinearity on the error sequence driving the algorithm. A nonlinear difference equation for the mean norm is explicitly solved using a differential equation approximation and integration by quadratures. The steady-state second-moment weight behavior is evaluated exactly for the erf nonlinearity. Using the above results, the tradeoff between the extent of error saturation, steady-state excess mean-square error, and rate of algorithm convergence is studied. The tradeoff shows that (1) starting with a sign detector, the convergence rate is increased by nearly a factor of two for each additional bit, and (2) as the number of bits is increased further, the additional bit by very little in convergence speed, asymptotically approaching the behavior of the linear algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Incremental Describing Function Analysis of Subharmonic Oscillations in Control Systems with Thyristor Converters

Often subharmonic oscillations of firing angle of thyristor converters with large magnitude in negative feedback control systems are a kind of jump phenomena. For their investigation rectifiers may be modeled by an asymmetrical saturation type nonlinearity in series with a linear block whose factors depend upon the number of pulses of the rectifier and the order of subharmonic in question. Application of the incremental describing function technique results in a set of curves, which is useful for the determination of the critical gain of the system and its design. Results of digital computer simulated systems are in good agreement with the predicted values

A Technique for Control of Traffic at Critical Intersections

An algorithm for control of queue lengths at oversaturated intersections oriented towards delaying spillback across upstream intersections as long as possible is described. It is based on determining the change in queue and the allotment of available green time so as to keep the ratio of the actual queues to the maximum link storage space on both phases equal. The mathematical development of the algorithm is described in terms of a Z transform analysis, and the conditions for stability are derived. The resulting mathematical model is linear with the exception of a saturation type nonlinearity caused by the physical requirement to limit split. A describing function analysis is used to demonstrate loop stability in the presence of the nonlinearity.