Further Improvements in High Gain Observer Design for Systems with Structured Nonlinearities

Due to their simple implementation based on a constant gain matrix, high gain observers are very common in practical applications. We consider systems whose dynamics can be decomposed into a linear and a nonlinear part, where the nonlinear part meets some Lipschitz condition. In many cases there exists a finite bound on the maximum feasible Lipschitz constant for which the error dynamics can be stabilized. Necessary and in some sense sufficient conditions for this maximum Lipschitz constant are given in [1]. These results has been improved in [2,3] by taking the structure of the linear part into account. Having a system with one single nonlinearity, the results given in [2,3] are strict. If multiple nonlinearities occur, even this approach tends to be to conservative. In this case, one could additionally take the internal structure of the nonlinearities into account which leads to a larger set of systems for which convergence of the observer error can be guaranteed. Our new approach is based on an approximation of the structured singular value [4] which yields existence conditions in terms of linear matrix inequalities (LMIs). These LMIs may as well be used for the numerical computation of the observer gain. We demonstrate the advantage of our method on an example. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive fuzzy output feedback control for a single-link flexible robot manipulator driven DC motor via backstepping

In this paper, an adaptive fuzzy output feedback approach is proposed for a single-link robotic manipulator coupled to a brushed direct current (DC) motor with a nonrigid joint. The controller is designed to compensate for the nonlinear dynamics associated with the mechanical subsystem and the electrical subsystems while only requiring the measurements of link position. Using fuzzy logic systems to approximate the unknown nonlinearities, an adaptive fuzzy filter observer is designed to estimate the immeasurable states. By combining the adaptive backstepping and dynamic surface control (DSC) techniques, an adaptive fuzzy output feedback control approach is developed. Stability proof of the overall closed-loop system is given via the Lyapunov direct method. Three key advantages of our scheme are as follows: (i) the proposed adaptive fuzzy control approach does not require that all the states of the system be measured directly, (ii) the proposed control approach can solve the control problem of robotic manipulators with unknown nonlinear uncertainties, and (iii) the problem of “explosion of complexity” existing in the conventional backstepping control methods is avoided. The detailed simulation results are provided to demonstrate the effectiveness of the proposed controller.     

Asymptotic analysis of a coupled nonlinear parabolic system

This paper deals with asymptotic analysis of a parabolic system with inner absorptions and coupled nonlinear boundary fluxes. Three simultaneous blow-up rates are established under different dominations of nonlinearities, and simply represented in a characteristic algebraic system introduced for the problem. In particular, it is observed that two of the multiple blow-up rates are absorption-related. This is substantially different from those for nonlinear parabolic problems with absorptions in all the previous literature, where the blow-up rates were known as absorptionindependent. The results of the paper rely on the scaling method with a complete classification for the nonlinear parameters of the model. The first example of absorption-related blow-up rates was recently proposed by the authors for a coupled parabolic system with mixed type nonlinearities. The present paper shows that the newly observed phenomena of absorptionrelated blow-up rates should be due to the coupling mechanism, rather than the mixed type nonlinearities.      

A finite element model for nonlinear behaviour of piezoceramics under weak electric fields

It has been experimentally observed that piezoceramic materials exhibit different types of nonlinearities under different combinations of electric and mechanical fields. When excited near resonance in the presence of weak e to a Duffinor such as jump phenomena and presence of superharmonics in the response spectra. There has not been much work in the litrature to model these types of nonlinearities. Some authors have developed one-dimensional models for the above phenomenon and derived closed-form solutions for the displacement response of piezo-actuators. However, the generalized three-dimensional (3-D) formulation of electric enthalpy, the variational formulation and the FEM implementation have not yet been addressed, which are the focus of this paper. In this work, these nonlinearities have been modelled in a 3-D piezoelectric continuum using higher order quadratic and cubic terms in the generalized electric enthalpy density function. The coupled nonlinear finite element equations have been derived using variational formulation. A special linearization technique for assembling the nonlinear matrices and solution of the resulting nonlinear equations has been developed. The method has been used for simulating the nonlinear frequency response of a lead zirconate titanate plate excited near its first in-plane vibration resonance frequency with sinusoidal excitations of different electric field strengths. The results have been compared with those of the experiment.     

Efficient approximations of univariate nonlinearities for linear planning models

Univariate nonlinearities occur either if the variables or parameters of a planning model are nonlinear functions of time, or if linear functions of time are multiplied by each other. Moreover, cost functions or revenue functions consist in most cases of univariate nonlinear terms. These nonlinearities are approximated by several types of piecewise-linear functions either particularly or simultaneously. In the case of linearizing nonlinearities in the decision variables it should be mentioned that the results of the approximation can only be used for linear models, if the convexity of the set of feasible solutions is guaranteed. Since the approximating procedure is based on variable nodes, the fit may be improved by optimizing the positions of these nodes. Therefore, this approach yields a far better fit than an approximation on the basis of equidistant and fixed nodes. As the number of variables and restrictions in linearized models increases according to the number of approximating intervals, the number of nodes determines substantially the size of a LP-model and therefore the expenses of a planning approach. Using the procedure described here, the number of approximating intervals will be minimal at a given goodness of fit and the model's size may be effectively reduced. On the other hand, for a certain number of intervals the most efficient approximation is found.     

Analytical solution in 2D domain for nonlinear response of piezoelectric slabs under weak electric fields

Piezoceramic materials exhibit different types of nonlinearities depending upon the magnitude of the mechanical and electric field strength in the continuum. Some of the nonlinearities observed under weak electric fields are: presence of superharmonics in the response spectra and jump phenomena etc. especially if the system is excited near resonance. In this paper, an analytical solution (in 2D plane stress domain) for the nonlinear response of a rectangular piezoceramic slab has been obtained by use of Rayleigh–Ritz method and perturbation technique. The eigenfunction obtained from solution of the differential equation of the linear problem has been used as the shape function in the Rayleigh–Ritz method. Forced vibration experiments have been conducted on a rectangular piezoceramic slab by applying varying electric field strengths across the thickness and the results have been compared with those of analytical solution. The analytical solutions compare well with those of experimental results. These solutions should serve as a method to validate the FE formulations as well as help in the determination of nonlinear material property coefficients for these materials.     

Bond graph model of an induction machine with hysteresis nonlinearities

An induction machine is one of the most convenient devices for conversion of electrical energy to mechanical rotational energy. Induction machine is a typical member of a multi-domain, nonlinear, high-order dynamic system. To reduce its complexity, the mathematical models used for designing their control have several assumptions built into them. The most striking of these assumptions is that of linear magnetics. Bond graph is a convenient tool for modelling nonlinear elements. This paper proposes the use of the bond graph methodology to develop a model of an induction machine that includes the nonlinearities due to magnetics.     

Nonlinearly coupled in-plane and transverse vibrations of a spinning disk

Previous nonlinear spinning disk models neglected the in-plane inertia of the disk since this permits the use of a stress function. This paper aims to consider the effect of including the in-plane inertia of the disk on the resulting nonlinear dynamics and to construct approximate solutions that capture the new dynamics. The inclusion of the in-plane inertia results in a nonlinear coupling between the in-plane and transverse vibrations of the spinning disk. The full nonlinear partial differential equations are simplified to a simpler nonlinear two degrees of freedom model via the method of Galerkin. A canonical perturbation approach is used to derive an approximate solution to this simpler nonlinear problem. Numerical simulations are used to evaluate the effectiveness of the approximate solution. Through the use of these analytical and numerical tools, it becomes apparent that the inclusion of in-plane inertia gives rise to new phenomena such as internal resonance and the possibility of instability in the system that are not predicted if the in-plane inertia is ignored. It is also demonstrated that the canonical perturbation approach can be used to produce an effective approximate solution.     

两自由度非对称三次系统非奇异时的非线性模态及叠加性

本文利用非线性模态子空间的不变性研究两自由度非对称三次系统在非奇异条件下的非线性模态及其模态叠加解有效性，重点考虑这种有效性与模态动力学方程静态分岔之间的关系·大量的数值结果表明，非线性模态解的有效性不仅与其局部性的限制有关，而且与模态动力学方程静态解分岔有关·     

On the Lipschitz stability of complex systems of differential equations

Mathematical models of real physical processes and phenomena can be described by the so-called complex systems of differential equations. Among them are systems that decompose into interacting subsystems and intrasystemic connections. They have significant nonlinearities. In this connection the application of the existing methods of study of the dynamics of the behavior of such systems is quite difficult. We propose some approaches to the study of Lipschitz stability connected with the technique of applying Lyapunov functions. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 12–16.     

Exact multiplicity results for variational inequalities involving pointwise constraints on the derivatives

This paper concerns a class of nonlinear variational problems involving pointwise constraints on the second derivatives. Our aim is to describe the set of data for which these problems have solutions and to analyse the structure of the set of solutions under suitable assumptions on the asymptotic behaviour of the nonlinear term. In particular, if this term is assumed to be convex, then we can specify the number of solutions and obtain exact multiplicity results.The existence, nonexistence and multiplicity results we obtain show that the presence of constraints of this kind produces some phenomena which are typical of nonlinear elliptic equations with “jumping” nonlinearities.     

Nonlinear phenomena in the dynamics of micromechanical gyroscopes

LL- and TT-type vibratory micromechanical gyroscopes (MMG) are considered with regard for nonlinear dependence of the suspension resistance forces and electrostatic forces on the displacement of the MMG sensitive elements. Nonlinear differential equations for a MMG operating in the measuring mode are obtained. These equations contain both analytical and nonanalytical nonlinearities. The effect of these nonlinearities on the dynamics and precision of vibratory MMGs is studied. The use of the method of averaging revealed stable steady-state modes of vibratory MMGs. The corresponding resonance curves are constructed. The results obtained may find application in the design of devices of the types considered.     

Filtering for a class of nonlinear discrete-time stochastic systems with state delays

In this paper, the filtering problem is investigated for a class of nonlinear discrete-time stochastic systems with state delays. We aim at designing a full-order filter such that the dynamics of the estimation error is guaranteed to be stochastically, exponentially, ultimately bounded in the mean square, for all admissible nonlinearities and time delays. First, an algebraic matrix inequality approach is developed to deal with the filter analysis problem, and sufficient conditions are derived for the existence of the desired filters. Then, based on the generalized inverse theory, the filter design problem is tackled and a set of the desired filters is explicitly characterized. A simulation example is provided to demonstrate the usefulness of the proposed design method.     

A Finite Element Formulation for the Nonlinear Analysis of Piezoelectric Three-Dimensional Beam Structures

A finite element formulation for a three-dimensional piezoelectric beam which includes geometrical and material nonlinearities is presented. To account for the piezoelectric effect, the coupling between the mechanical stress and the electrical displacement is considered. Based on the Timoshenko theory, an eccentric beam formulation is introduced which provides an efficient model to analyze piezoelectric structures. The geometrically nonlinear assumption allows the calculation of large deformations including buckling analysis. A quadratic approximation of the electric potential through the cross section of the beam ensures the fulfilment of the charge conservation law exactly. This assumption leads to a finite element formulation with six mechanical and five electrical degrees of freedom per node. To take into account the typical ferroelectric hysteresis phenomena, a nonlinear material model is essential. For this purpose, the phenomenological Preisach model is implemented into the beam formulation which provides an efficient determination of the remanent part of the polarization. The applicability of the introduced beam formulation is discussed with respect to available data from literature. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic stability of ground states in 3D nonlinear Schrödinger equation including subcritical cases

We consider a class of nonlinear Schrödinger equation in three space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L²) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a careful chosen one parameter family of bound states that “shadows” the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations.     

Effects of higher nonlinearity on the dynamics and synchronization of two coupled electromechanical devices

The dynamics and synchronization of coupled electromechanical systems with both cubic and quintic nonlinearities are analyzed. A detail attention is carried out to the study of the effects of the introduced quintic nonlinearity on the amplitudes of the harmonic oscillatory states, the stability boundaries of the harmonic oscillations, and on the bifurcation structures. We examine the synchronization phenomena on the unidirectional capacitive and resistive coupled such electromechanical systems both in their regular and chaotic states. The stability of synchronization process is studied follows the Floquet theory and Hill infinite determinant. Numerical simulations confirm and complement the results obtained by the analytical approach.     

Multi-pulse chaotic motions of a rotor-active magnetic bearing system with time-varying stiffness

In this paper, we investigate the Shilnikov type multi-pulse chaotic dynamics for a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time-varying in a periodic form. The dimensionless equation of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions is a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found from the numerical results that there are the phenomena of the Shilnikov type multi-pulse chaotic motions for the rotor-AMB system. A new jumping phenomenon is discovered in the rotor-AMB system with the time-varying stiffness.     

多自由度强非线性颤振分析的增量谐波平衡法

对多个自由度上含有强非线性项系统的颤振问题,推广应用增量谐波平衡法进行分析．考虑带有强非线性立方平移和俯仰刚度项的二元机翼颤振方程,首先将方程用矩阵形式表示,然后把振动过程分解成为振动瞬态的持续增量过程,再采用振幅作为控制参数应用谐波平衡法,以这种推广的增量谐波平衡法求得方程解的表达式,并由此分析系统的分岔现象、极限环颤振现象和谐波项数的取值问题,最后用龙格-库塔数值方法进行验算,结果表明:分析多个自由度的强非线性颤振,增量谐波平衡法是精确有效的．     

Asymptotic stability of ground states in 2D nonlinear Schrödinger equation including subcritical cases

We consider a class of nonlinear Schrödinger equations in two space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L²) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time-dependent, Hamiltonian, linearized dynamics around a carefully chosen one-parameter family of bound states that “shadows” the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations.     

Equation admitting linearization and describing waves in dissipative media with modular,quadratic, and quadratically cubic nonlinearities

A second-order partial differential equation admitting exact linearization is discussed. It contains terms with nonlinearities of three types—modular, quadratic, and quadratically cubic—which can be present jointly or separately. The model describes nonlinear phenomena, some of which have been studied, while others call for further consideration. As an example, individual manifestations of modular nonlinearity are discussed. They lead to the formation of singularities of two types, namely, discontinuities in a function and discontinuities in its derivative, which are eliminated by dissipative smoothing. The dynamics of shock fronts is studied. The collision of two single pulses of different polarity is described. The process reveals new properties other than those of elastic collisions of conservative solitons and inelastic collisions of dissipative shock waves.