Some new concepts in nonlinear systems

Some new concepts characterizing the response of nonlinear systems are developed. These new concepts are denoted by the terms, the transient system equivalent, the response vector, and the space-phase components. This third concept is analyzed in comparison with the well-known technique of symmetrical components. The performance of a multiplicative feedback control system is represented by a nonlinear integro-differential equation; its solution is obtained by the principle of variation of parameters. The system response is treated as a vector and is resolved into its space-phase components. The individual effects of these components on the performance of the system are discussed. The suitability of the technique for the transient analysis of higher order nonlinear control systems is discussed.

Application of D transforms in the analysis of nonlinear distributed parameter systems

The scope of application of Laplace transforms presently limited to the study of linear partial differential equations, is extended to the nonlinear domain by this study. This has been achieved by modifying the definition of D transforms, put forth recently for the study of classes of nonlinear lumped parameter systems. The appropriate properties of the new D transforms are presented to bring out their applicability in the analysis of nonlinear distributed parameter systems.

L2-Stability Of A Class Of Nonlinear-Systems

Sufficient conditions are given for the L2-stability of a class of feedback systems consisting of a linear operator G and a nonlinear gain function, either odd monotone or restricted by a power-law, in cascade, in a negative feedback loop. The criterion takes the form of a frequency-domain inequality, Re[1 + Z(jω)] G(jω) δ > 0 ω ε (−∞, +∞), where Z(jω) is given by, Z(jω) = β[Y1(jω) + Y2(jω)] + (1 − β)[Y3(jω) − Y3(−jω)], with 0 β 1 and the functions y1(·), y2(·) and y3(·) satisfying the time-domain inequalities, ∝−∞+∞¦y1(t) + y2(t)¦ dt 1 − ε, y1(·) = 0, t < 0, y2(·) = 0, t > 0 and ε > 0, and , c2 being a constant depending on the order of the power-law restricting the nonlinear function. The criterion is derived using Zames' passive operator theory and is shown to be more general than the existing criteria

Nonlinear aeroelasticity of rotating and flapping wings - a review

The interaction between large deflections, rotation effects and unsteady aerodynamics makes the dynamic analysis of rotating and flapping wing a nonlinear aeroelastic problem. This problem is governed by nonlinear periodic partial differential equations whose solution is needed to calculate the response and loads acting on vehicles using rotary or flapping wings for lift generation. We look at three important problems in this paper. The first problem shows the effect of nonlinear phenomenon coming from piezoelectric actuators used for helicopter vibration control. The second problem looks at the propagation on material uncertainty on the nonlinear response, vibration and aeroelastic stability of a composite helicopter rotor. The third problem considers the use of piezoelectric actuators for generating large motions in a dragonfly inspired flapping wing. These problems provide interesting insights into nonlinear aeroelasticity and show the likelihood of surprising phenomenon which needs to be considered during the design of rotary and flapping wing vehicle

Nonlinear vibration analysis of composite laminated and sandwich plates with random material properties

Nonlinear vibration analysis is performed using a C-0 assumed strain interpolated finite element plate model based on Reddy's third order theory. An earlier model is modified to include the effect of transverse shear variation along the plate thickness and Von-Karman nonlinear strain terms. Monte Carlo Simulation with Latin Hypercube Sampling technique is used to obtain the variance of linear and nonlinear natural frequencies of the plate due to randomness in its material properties. Numerical results are obtained for composite plates with different aspect ratio, stacking sequence and oscillation amplitude ratio. The numerical results are validated with the available literature. It is found that the nonlinear frequencies show increasing non-Gaussian probability density function with increasing amplitude of vibration and show dual peaks at high amplitude ratios. This chaotic nature of the dispersion of nonlinear eigenvalues is also r

Response of nonlinear systems to narrow-band excitation

The hardening cubic spring oscillator is studied under narrow-band gaussian excitation. Equivalent linearization leads to multiple steady states. The realizability of the solution is discussed through stochastic stability analysis. Theoretical results are supported by numerical simulation.

Performance analysis of a suprathreshold, stochastic resonance based nonlinear detector

In this paper we introduce a nonlinear detector based on the phenomenon of suprathreshold stochastic resonance (SSR). We first present a model (an array of 1-bit quantizers) that demonstrates the SSR phenomenon. We then use this as a pre-processor to the conventional matched filter. We employ the Neyman-Pearson(NP) detection strategy and compare the performances of the matched filter, the SSR-based detector and the optimal detector. Although the proposed detector is non-optimal, for non-Gaussian noises with heavy tails (leptokurtic) it shows better performance than the matched filter. In situations where the noise is known to be leptokurtic without the availability of the exact knowledge of its distribution, the proposed detector turns out to be a better choice than the matched filter.

An Empirical Comparison of Linear and Nonlinear Volatility Models for Nordic Stock Returns

This paper examines how volatility in financial markets can preferable be modeled. The examination investigates how good the models for the volatility, both linear and nonlinear, are in absorbing skewness and kurtosis. The examination is done on the Nordic stock markets, including Finland, Sweden, Norway and Denmark. Different linear and nonlinear models are applied, and the results indicates that a linear model can almost always be used for modeling the series under investigation, even though nonlinear models performs slightly better in some cases. These results indicate that the markets under study are exposed to asymmetric patterns only to a certain degree. Negative shocks generally have a more prominent effect on the markets, but these effects are not really strong. However, in terms of absorbing skewness and kurtosis, nonlinear models outperform linear ones.

Dynamic programming based optimum non-uniform samples for speech reconstruction and coding

Non-uniform sampling of a signal is formulated as an optimization problem which minimizes the reconstruction signal error. Dynamic programming (DP) has been used to solve this problem efficiently for a finite duration signal. Further, the optimum samples are quantized to realize a speech coder. The quantizer and the DP based optimum search for non-uniform samples (DP-NUS) can be combined in a closed-loop manner, which provides distinct advantage over the open-loop formulation. The DP-NUS formulation provides a useful control over the trade-off between bitrate and performance (reconstruction error). It is shown that 5-10 dB SNR improvement is possible using DP-NUS compared to extrema sampling approach. In addition, the close-loop DP-NUS gives a 4-5 dB improvement in reconstruction error.

A generalized Linear Programming based approach to optimal divisible load scheduling

In this paper we propose a general Linear Programming (LP) based formulation and solution methodology for obtaining optimal solution to the load distribution problem in divisible load scheduling. We exploit the power of the versatile LP formulation to propose algorithms that yield exact solutions to several very general load distribution problems for which either no solutions or only heuristic solutions were available. We consider both star (single-level tree) networks and linear daisy chain networks, having processors equipped with front-ends, that form the generic models for several important network topologies. We consider arbitrary processing node availability or release times and general models for communication delays and computation time that account for constant overheads such as start up times in communication and computation. The optimality of the LP based algorithms is proved rigorously.

Robust/optimal temperature profile control using neural networks

An approximate dynamic programming (ADP) based neurocontroller is developed for a heat transfer application. Heat transfer problem for a fin in a car's electronic module is modeled as a nonlinear distributed parameter (infinite-dimensional) system by taking into account heat loss and generation due to conduction, convection and radiation. A low-order, finite-dimensional lumped parameter model for this problem is obtained by using Galerkin projection and basis functions designed through the 'Proper Orthogonal Decomposition' technique (POD) and the 'snap-shot' solutions. A suboptimal neurocontroller is obtained with a single-network-adaptive-critic (SNAC). Further contribution of this paper is to develop an online robust controller to account for unmodeled dynamics and parametric uncertainties. A weight update rule is presented that guarantees boundedness of the weights and eliminates the need for persistence of excitation (PE) condition to be satisfied. Since, the ADP and neural network based controllers are of fairly general structure, they appear to have the potential to be controller synthesis tools for nonlinear distributed parameter systems especially where it is difficult to obtain an accurate model.

Effective treatment of infectious diseases: A nonlinear adaptive control theoretic approach

A nonlinear adaptive system theoretic approach is presented in this paper for effective treatment of infectious diseases that affect various organs of the human body. The generic model used does not represent any specific disease. However, it mimics the generic immunological dynamics of the human body under pathological attack, including the response to external drugs. From a system theoretic point of view, drugs can be interpreted as control inputs. Assuming a set of nominal parameters in the mathematical model, first a nonlinear controller is designed based on the principle of dynamic inversion. This treatment strategy was found to be effective in completely curing "nominal patients". However, in some cases it is ineffective in curing "realistic patients". This leads to serious (sometimes fatal) damage to the affected organ. To make the drug dosage design more effective, a model-following neuro-adaptive control design is carried out using neural networks, which are trained (adapted) online. From simulation studies, this adaptive controller is found to be effective in killing the invading microbes and healing the damaged organ even in the presence of parameter uncertainties and continuing pathogen attack.