A simple model‐based nonlinear control strategy for feedback linearizable nonlinear processes

Abstract

In this paper we propose a simple and novel model‐based nonlinear control strategy for feedback linearizable nonlinear processes. The nonlinear controller, called NLC, has a forward static nonlinear element and, if the relative degree of the process is greater than unity, a linear dynamic feedback path. The nonlinear static element is a modified hyperbolic function which involves two tunable parameters used to adjust the linear gain and saturation level of the NLC. The NLC parameters are determined to achieve a desired tracking transfer function of the form (K_cs + 1)/(1 + ?s) ^r+1 where K_c and ? are parameters relating to NLC controller settings and r the system's relative degree. The effectiveness and applicability of the proposed nonlinear control strategy are demonstrated through the controls of nonlinear chemical processes. Extensive simulation results show that the proposed nonlinear control strategy is a robust model‐based approach for the nonlinear control of dynamic processes.     

Intense low threshold nonlinear absorption and nonlinear refraction in a new organic–polymer nanocomposite

Intense reverse saturable absorption is reported for the first time in solid films of a new organic–polymer nanocomposite, cast by doping Biebrich Scarlet dye in a vinyl polymer host polyvinyl alcohol for various concentrations, as studied employing the Z-scan technique at 442 nm under different peak incident intensities ranging from 9.37 × 10² to 104.18 × 10² W cm⁻². The sample also exhibited nonlinear refraction under the experimental conditions. The estimated values of the effective coefficients of nonlinear absorption β_eff(0.27 × 10⁻² to 45.5 × 10⁻² cm W⁻¹) as well as nonlinear refraction n₂ (−1.5 × 10⁻⁷ to −2.75 × 10⁻⁷ cm² W⁻¹) measured up to the highest reported ones for low power continuous wave excitation. The composite films were characterized as nanoclusters consisting of dye molecules encapsulated between larger molecules of the amorphous polymer and having a low average roughness (≈1 nm) for the surface. These results, together with the simple and flexible processing method for the dye–polymer composite, imply that BS–PVA composite films have promising optical properties as an efficient low threshold nanocomposite material for potential applications in nonlinear optical devices.     

On perfect nonlinear functions (Π)

Perfect nonlinear functions are of importance in cryptography. By using Galois ring, relative trace and investigating the character values of corresponding relative difference sets, we present a construction of perfect nonlinear functions from to , where m′ is a divisor of 2m, and a construction of perfect nonlinear functions from to where 2m is possibly larger than the largest divisor of n. Meanwhile we prove that there exists a perfect nonlinear function from to if and only if p = 2, and there doesn’t exist a perfect nonlinear function from to if m > n and l(l is odd) is self-conjugate modulo 2 ^k (k ≥ 1).      

A nonlinear viscoelastic–viscoplastic model for adhesives

We consider the nonlinear viscoelastic–viscoplastic behavior of adhesives. We develop a one-dimensional nonlinear model by combining Schapery’s nonlinear single integral model and Perzyna’s viscoplastic model. The viscoplastic strain was solved iteratively using the von Mises yield criterion and nonlinear kinematic hardening. The combined viscoelastic–viscoplastic model was solved using Newton’s iteration and implemented into a finite element model. The model was calibrated using creep-recovery data from bulk adhesives and verified from the cyclic behavior of the bulk adhesives. The finite element model results agreed with experimental creep and cyclic responses, including recoverable and permanent strain after load removal. Although the contribution of the viscoplastic strain was small, both viscoplastic and viscoelastic components of strain response were required to describe the adhesive creep and cyclic response.

     

Similarity solutions for nonlinear diffusion — a new integration procedure

The nonlinear diffusion equation arises in many important areas of science and technology and most of the known exact solutions turn out to be similarity solutions. For a general similarity solution involving an arbitrary parameter , a new integration procedure is proposed which enables first integrals to be obtained for special values of . The best known exact solutions arise from this analysis when the integration constant is taken to be zero and the procedure provides a natural way of deducing other special exact solutions. A new exact solution is obtained for the power law diffusivity of index —4/3 and new first integrals are deduced for a general equation which includes nonlinear cylindrical and spherical symmetrical diffusion and one-dimensional nonlinear diffusion with an inhomogeneous diffusivity. The procedure has given rise to an extensive number of first-order ordinary differential equations which include a wide variety of differing physical situations and which warrant further study either analytically to determine exact integrals or numerically for particular boundary value problems.     

Time evolution of three-dimensional nonlinear gravity–capillary free-surface flows

The evolution in time of three-dimensional gravity and gravity–capillary free-surface flows generated by a moving pressure distribution is considered. Solutions of the fully nonlinear equations in deep water are calculated by boundary-integral-equation methods and marching in time. Comparisons between unsteady and steady solutions are discussed.     

Long-wave transverse instability of weakly nonlinear gravity–capillary solitary waves

The long-wave transverse instability of weakly nonlinear gravity–capillary solitary waves is verified from the weakly nonlinear cubic-order truncation model derived from the free-surface boundary conditions of the Euler equations in the water-wave problem. The linearized operators corresponding to the cubic-order truncation model feature a skew-symmetric structure, consistent to the associated property of the Euler equations. From this, the leading-order initial long-wave transverse instability growth rate of the weakly nonlinear gravity–capillary solitary waves is estimated to be quantitatively identical, in the weakly nonlinear limit, to the earlier result obtained from the full Euler equations, through an equivalent perturbation procedure.     

Unit commitment in oligopolistic markets by nonlinear mixed variable programming

In the paper we consider the unit commitment problem in oligopolistic markets. The formulation of the problem involves both integer and continuous variables and nonlinear functions as well, thus yielding a nonlinear mixed variable programming problem. Our formulation takes into account all technical constraints for the generating units, such as ramp rate and minimum up and down time constraints, considers the uncertainty related to the selling prices and allows modeling their dependence on the total output of a producer. The objective function is the expected value of the revenue over the different scenarios minus a term which takes into account the risk related to the decision. To solve the problem we adopt a recently proposed method for mixed integer nonlinear programming problems and use a derivative free algorithm to solve the continuous subproblems. We report results for two operators: one managing a single unit and the other managing three units. Numerical results give evidence to the features of the modeling and show viability of the adopted algorithm.     

A behavior based nonlinear multi‐agent framework using a hybrid approach

Abstract

In this paper, a novel design and implementation of a multi‐agent system via a hybrid approach is proposed. First of all, an autonomous agent is insured to be stable by employing the proposed local feedback control besides appropriate switching among well‐posed behaviors including obstacle avoidance, target approaching, wandering, manipulator grasping, etc. Next, a hybrid framework of the multi‐agent system (HMAS) is established for dealing with multi‐agent cooperative problems. The architecture of the inherent hierarchical control is designed consisting of two levels, where the first (lower) level mainly handles the continuous state describing the data acquired from various sensors, whereas the second (higher) level is responsible for providing a well performed supervisory control that can manipulate all available resources to successfully accomplish the underlying mission. In the Advanced Control Laboratory of the Department of Electrical Engineering at National Taiwan University, an actual wheeled mobile robot (WMR) has been developed, named The Treasure Hunter (TTH), which is able to hunt treasures in an unknown environment whereby the effectiveness of the proposed approach can be satisfactorily demonstrated.     

Analytical study of solitons to Biswas–Milovic model in nonlinear optics

Optical solitions to the Biswas–Milovic model have been investigated analytically with the help of the generalized Kudryashov method. Four types of nonlinear are considered. They are Kerr law, Power law, parabolic law and dual-power law. As a result, exact solitary wave solutions are obtained.     

Higher order dispersion effects in the noninstantaneous nonlinear Schrödinger equation

We present a systematic analysis of the effects, of higher-order dispersion, noninstantaneous nonlinear response, as well as stochastic coefficients in optical fiber. This study is motivated by recent experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber. Analytical expression of pulse amplitude is deduced with the second-order gain nonuniformity and the stimulated-Brillouin scattering-induced third-order as well as fourth-order dispersion effects involved. The influence of stochasticity, as well as the delayed Raman response in the nonconventional sidebands obtained due to the fourth-order dispersion, is considered. We note that the shape of the spectrum, and in particular the relative intensities of the higher order harmonics, is highly sensitive to the initial presence of classical noise, and can therefore be taken as a signature that the MI is seeded by vacuum fluctuations. Some direct simulations to see the evolution of different continuous wave states are reported. These show the formation of modulation instability pulses as well as transitions from lower amplitude continuous wave states to higher amplitude continuous wave states. The present results fit well with recent experimental investigations.     

A theoretical study of nonlinear magnetoelectric effect in magnetostrictive–piezoelectric trilayer

A novel nonlinear theoretical model is established for magnetoelectric (ME) effect in trilayer of magnetostrictive and piezoelectric phases, in which the nonlinear magnetic–mechanical coupling behavior for the magnetostrictive phase is firstly taken into account. In this theoretical model, the interface coupling parameter k is used for characterizing actual bonding conditions at the interface. The coupled magnetic–mechanical–electric effect involving linear and nonlinear coupling interactions in the ME laminated composites is numerically simulated using this nonlinear model. The numerical results predict giant ME effect for Terfenol-D based ME laminated composites. The quantitative dependences of the giant ME effect on the applied magnetic field, the piezoelectric property of piezoelectric phase, the volume fraction of magnetostrictive phase and the interface coupling parameter k are discussed in details. All of these dependences indicate that the nonlinear theoretical model established in this article can accurately capture nonlinear magnetic–mechanical–electric coupling behavior for Terfenol-D based ME laminated composites. The giant ME effect predicted for the Terfenol-D/PMN-PT/Terfenol-D composites is in excellent agreement with recent experimental data available. It confirms the validity and reliability of the obtained nonlinear theoretical model, and demonstrates the significance and necessity of considering the nonlinear magnetic–mechanical coupling behavior of Terfenol-D.     

Unsteady state shnulation of “SORBEX” system with nonlinear adsorption isotherms

An unsteady-state model of the “SORBEX” system is developed using a nonlinear equilibrium isotherm. This model is solved from the unsteady state to steady state using the dispersed plug flow model with Danckwert boundary conditions. The effect of various process parameters on the performance of the system are investigated. The present study reveals that the system performance and dynamics are strongly dependent on bed length, column diameter, feed and eluent flow rate, switch time, and nonlinearity of the isotherm. There exists a set of optimum values of these process parameters that represents the proper combination for best performance. The results obtained from the present simulation for the limiting case of linear isotherms (glucose-fructose) and nonlinear isotherms (monoethanol-amine-methanol) systems are compared with experimental data and were found to agree well.     

Optical solitons in a new type of coupled nonlinear Schrödinger equations

Abstract

We consider a new type of integrable coupled nonlinear Schrödinger (CNLS) equations proposed by Park and Shin [1999, Phys. Rev. E, 59, 2372]. Using the Ablowitz-Kaup-Newell-Segur method, we construct the Lax pair for simultaneous propagation of four fields in the new type of CNLS equations. The explicit form of soliton solutions are derived using the Hirota bilinear method.     

Stability criteria for neutral systems with time‐varying delays and nonlinear uncertainties

Abstract

The asymptotic stability problem for a class of neutral systems with time‐varying delays and nonlinear uncertainties is investigated in this paper. LMI‐based delay‐dependent criteria are proposed to guarantee the asymptotic stability of the considered systems. New Lyapunov‐Krasovskii functional and Leibniz‐Newton formulae are used to find the delay‐dependent stability results. Finally, some numerical examples are illustrated to show the improved results from using this method.     

Synthesis,growth and characterization of π conjugated organic nonlinear optical chalcone derivative

A new potentially useful nonlinear optical organic material, 1-(5-chlorothiophen-2-yl)-3-(2,3-dimethoxyphenyl)prop-2-en-1-one, has been synthesized and grown as a high-quality single crystal by the slow evaporation technique. The grown crystals were characterized by FT-IR, NMR, thermal analysis, and UV–visible spectroscopy. The material is thermally stabile up to 111 °C. The mechanical property of the grown crystals was studied using Vickers microhardness tester and the load dependence hardness was observed. The third order nonlinear optical properties of the material such as real and imaginary part of χ⁽³⁾, nonlinear absorption coefficient and nonlinear refractive index were determined using nanosecond laser pulses at 532 nm wavelength by employing Z-scan technique. The nonlinear refractive index is found to be of the order of 10⁻¹¹ cm² W⁻¹. The magnitude of third order susceptibility is of the order of 10⁻¹³ esu. The observed increase in the third order nonlinearity in these molecules clearly indicates the electronic origin. The compounds exhibit good optical limiting at 532 nm. The best optical limiting behavior of this molecule is due to the substituted strong electron donor.     

A microscopic nonlinear programming approach to shakedown analysis of cohesive–frictional composites

A microscopic approach together with nonlinear programming technique and finite element method is developed for shakedown analysis of a composite which has cohesive–frictional constituents. The macroscopic shakedown limit of a composite subject to cyclic loading is calculated in a direct way and the macro–micro relation is quantitatively evaluated. First, by means of the homogenization theory, the classical kinematic theorem of shakedown analysis is generalized to incorporate the microstructure – Representative Volume Element (RVE) chosen from a periodic heterogeneous material. Pressure-dependence and non-associated plastic flow of cohesive–frictional constituent materials are formulated into shakedown analysis. Based on the mathematical programming technique and the finite element method, the numerical micro-shakedown model is finally formulated as a nonlinear programming problem subject to only a few equality constraints, which is solved by a generalized Lagrangian-penalty iterative algorithm. The proposed approach provides a direct approach for determining the reduced macroscopic strength domain of heterogeneous or composite materials due to cyclic loading. Meanwhile, it can capture different plastic behaviors of materials and therefore the developed method has a wide applicability.     

Simulations of nonlinear fracture process zone in cementitious material—a boundary element approach

A boundary element method based numerical model is presented to simulate the nonlinear fracture process zone in cementitious materials. A cohesive type stress-separation constitutive relationship (-w curve) is incorporated in the model to represent the process zone. Numerical algorithms for both force-controlled (prescribed loading history) and crack-tip-control-led (prescribed crack tip position) are implemented to allow whole range simulations, including strain-softening and snap-back behavior. The numerical program includes special features to permit re-adjustment of nodal points such that accurate determination of the crack-tip position is achieved. A series of numerical simulations on both 3-point beams and double cantilever beams (DCB) are conducted to investigate the development of the inelastic process zone with respect to load level, loading configuration, specimen size, and the stress-separation relationship in the process zone. Size effect on fracture resistance is clearly demonstrated. Conclusions are drawn regarding the importance of determining the details of -w curve (i.e., the values of f _t and w _c )and the need for re-evaluating the R-curves approach in cementitious materials.     

Springback simulation of advanced high strength steels considering nonlinear elastic unloading–reloading behavior

The stress–strain response of some materials, such as advanced high strength steels, during unloading is nonlinear after the material has been loaded into the plastic deformation region. Upon reloading, the response shows a nonlinear elastic response that is different from that in unloading. Therefore, unloading–reloading of these materials forms a hysteresis loop in the elastic region. The Quasi-plastic–elastic model (Sun and Wagoner, 2011) was modified and combined with both isotropic-nonlinear kinematic hardening and two-surface plasticity models to simultaneously describe the nonlinear unloading response and complex cyclic response of sheet metals in the plastic region. The model was implemented as user-defined material subroutines, i.e. UMAT and VUMAT, for ABAQUS/Standard and ABAQUS/Explicit finite element codes, respectively. Uniaxial loading-unloading tests were performed on three common grades of automotive sheet steel: DP600, DP980 and TRIP780 steel. The model was verified by comparing the predicted material response with the corresponding experimental response. Finally, the model was used to predict the springback of a U-shape channel section formed in a plane-strain channel draw process. The results showed that the model was able to considerably improve springback predictions compared to the usual assumption of linear elastic unloading.