Nonlinear Systems Theory Research Articles

Excitation power quantities in phase resonance testing of nonlinear systems with phase-locked-loop excitation

Phase resonance testing is one method for the experimental extraction of nonlinear normal modes. This paper proposes a novel method for nonlinear phase resonance testing. Firstly, the issue of appropriate excitation is approached on the basis of excitation power considerations. Therefore, power quantities known from nonlinear systems theory in electrical engineering are transferred to nonlinear structural dynamics applications. A new power-based nonlinear mode indicator function is derived, which is generally applicable, reliable and easy to implement in experiments. Secondly, the tuning of the excitation phase is automated by the use of a Phase-Locked-Loop controller. This method provides a very user-friendly and fast way for obtaining the backbone curve. Furthermore, the method allows to exploit specific advantages of phase control such as the robustness for lightly damped systems and the stabilization of unstable branches of the frequency response. The reduced tuning time for the excitation makes the commonly used free-decay measurements for the extraction of backbone curves unnecessary. Instead, steady-state measurements for every point of the curve are obtained. In conjunction with the new mode indicator function, the correlation of every measured point with the associated nonlinear normal mode of the underlying conservative system can be evaluated. Moreover, it is shown that the analysis of the excitation power helps to locate sources of inaccuracies in the force appropriation process. The method is illustrated by a numerical example and its functionality in experiments is demonstrated on a benchmark beam structure.

Rendezvous of multiple nonholonomic unicycles-based on backstepping

ABSTRACTRendezvous problem of nonholonomic unicycles has been considered in this paper and nonlinear distributed controllers based on backstepping are constructed to guarantee the consensus of both position and direction simultaneously. Cascaded theory of nonlinear system is utilised to reduce the rendezvous problem of nonholonomic unicycles to consensus problem of moving direction and consensus problem of position, respectively. The uncontrollable challenge of position system after consensus of moving direction has been successfully removed through dilated coordinate transformation. Then, a modified backstepping methodology based on the coupling structure of position system is derived to guarantee the consensus of each unicycle's position. In the end, not only consensus of position but also consensus of direction has been realised for multiple nonholonomic unicycles using the controllers proposed in this paper. Simulation results using Matlab illustrate the effectiveness of the results of this paper.

Dissipativity Theory for Nonlinear Stochastic Dynamical Systems

In this paper, we develop stochastic dissipativity theory for nonlinear dynamical systems using basic input-output and state properties. Specifically, a stochastic version of dissipativity using both an input-output as well as a state dissipation inequality in expectation for controlled Markov diffusion processes is presented. The results are then used to derive extended Kalman–Yakubovich–Popov conditions for characterizing necessary and sufficient conditions for stochastic dissipativity of stochastic dynamical systems using two-times continuously differentiable storage functions. In addition, feedback interconnection stability in probability results for stochastic dynamical systems are developed thereby providing a generalization of the small gain and positivity theorems to stochastic systems.

Saleh’s Model of AM/AM and AM/PM Conversions is not a Model without Memory – Another Proof

Abstract The so-called Saleh’s representation for description of the AM/AM and AM/PM conversions, occurring in communication power amplifiers, consists of two expressions that describe them as functions of a real-valued baseband signal modulating the carrier amplitude. It is a common view that this description forms a model without memory. We show here that the above belief is not correct; just the opposite is true. To prove this, we take into account an equivalent description of the Saleh’s model called the quadrature model of bandpass nonlinearities and express it in a form of a nonlinear operator. Afterwards, we check whether this operator possesses a zero memory. To this end, we use an appropriate theorem of the nonlinear systems theory. Finally, as a result of this investigation, we observe that the memory of the above operator is nonzero.

A 0.92-THz SiGe Power Radiator Based on a Nonlinear Theory for Harmonic Generation

We propose a nonlinear device model and a systematic methodology to generate maximum power at any desired harmonic. The proposed power optimization technique is based on the Volterra-Wiener theory of nonlinear systems. By manipulating the device nonlinearity and optimizing the embedding network, optimum conditions for harmonic power generation are provided. Using this theory, a 920–944-GHz frequency quadrupler is designed in a 130-nm SiGe:C process. The circuit achieves the peak output power of −17.3 and −10 dBm of effective isotropic radiated power and consumes 5.7 mW of dc power. To the best of our knowledge, this circuit demonstrates the highest generated power among Si/SiGe-based sources at this frequency range.

Dynamical System Analysis of Interacting Hessence Dark Energy inf(T)Gravity

We have carried out dynamical system analysis of hessence field coupling with dark matter inf(T)gravity. We have analysed the critical points due to autonomous system. The resulting autonomous system is nonlinear. So, we have applied the theory of nonlinear dynamical system. We have noticed that very few papers are devoted to this kind of study. Maximum works in literature are done treating the dynamical system as done in linear dynamical analysis, which are unable to predict correct evolution. Our work is totally different from those kinds of works. We have used nonlinear dynamical system theory, developed till date, in our analysis. This approach gives totally different stable solutions, in contrast to what the linear analysis would have predicted. We have discussed the stability analysis in detail due to exponential potential through computational method in tabular form and analysed the evolution of the universe. Some plots are drawn to investigate the behaviour of the system(this plotting technique is different from usual phase plot and that devised by us). Interestingly, the analysis shows that the universe may resemble the “cosmological constant” like evolution (i.e.,ΛCDM model is a subset of the solution set). Also, all the fixed points of our model are able to avoid Big Rip singularity.

Fractional dynamical systems: A fresh view on the local qualitative theorems

The aim of this work is to describe the qualitative behavior of the solution set of a given system of fractional differential equations and limiting behavior of the dynamical system or flow defined by the system of fractional differential equations. In order to achieve this goal, it is first necessary to develop the local theory for fractional nonlinear systems. This is done by the extension of the local center manifold theorem, the stable manifold theorem and the Hartman-Grobman theorem to the scope of fractional differential systems. These latter two theorems establish that the qualitative behavior of the solution set of a nonlinear system of fractional differential equations near an equilibrium point is typically the same as the qualitative behavior of the solution set of the corresponding linearized system near the equilibrium point. Furthermore, we discuss the stability conditions for the equilibrium points of these systems. We point out that, the fractional derivative in these systems is in the Caputo sense.

Satisfactory fault tolerant control with soft-constraint for discrete time-varying systems: numerical recursive approach

The problem of active fault-tolerant control (FTC) with multiple performance constraints including the closed-loop stability, the control input and system state constraints, and the optimized quadratic performance as well, is investigated for a class of discrete time-varying systems subject to sensor faults. A satisfactory FTC strategy is proposed by means of the numerical recursion to guarantee the acceptable performance constraints on the faulty closed-loop system. In the strategy, according to the constraints relaxation technique from MPC technology, a soft-constraint based online FTC design method is developed to explicitly handle the multi-objective requirements of the performance constraints. Its main contributions include that during the online FTC controller reconfiguration, the infeasibility due to the possibly inconsistent constraints is avoided by relaxing soft constraints on the one hand, such that the continuous operation of the closed-loop FTC system is ensured. On the other hand, the developed nonlinear and non-convex optimization calculation for the numerical FTC controller design is converted into a convex quadratically constrained quadratic programming (QCQP) under certain condition, to solve the NP-hard problem due to the intrinsic non-convex constraints. Moreover, the stability of the closed-loop FTC system is analyzed based on the stability theory of discrete-time nonlinear system. Finally, two simulative examples are presented to verify the effectiveness of the proposed satisfactory FTC method.

Multiple solutions for the nonparametric Plateau problem within the Euclidean space $$\mathbb R^p$$ R p of arbitrary dimension

With the aid of Dirichlet’s principle, we can minimize the area-functional for graphs in \(\mathbb R^p\) defined on convex domains \(\varOmega \subset \mathbb R^2\); here the dimension \(p\ge 3\) is arbitrary. In the minimizing sequence we replace the graphs with embeddings of the unit disc via the uniformization method, and then we substitute them by the harmonic extensions of their boundary values. This is possible within the class of harmonic embeddings due to a beautiful result by Kneser (Jahresber Dt Math Ver 35:123–124, 1926). We calculate the first and second variation of the area-functional with higher codimensions explicitly. In the beginning we present the quasilinear nonparametric minimal surface system under Dirichlet boundary conditions, where uniqueness does not seem to prevail for \(p\ge 4\). The constructed solutions above are stable in the sense that they possess a nonnegative second variation. However, we indicate by an example how to construct minimal graphs which are unstable. Furthermore, we prove a mountain-pass lemma: an absolute minimizer together with a further strict relative minimizer of the area-functional possess a third minimal graph on the mountain-pass. In comparison with the Morse theory for Dirichlet’s integral developed by R. Courant, M. Shiffman, G. Strohmer, A. Tromba, M. Struwe, J. Jost, and R. Jakob, we can consider critical points of the area-functional and apply the continuity theorem of Morse and Tompkins (Am J Math 63:825–838, 1941). We refrain from the construction of certain minimizing paths, however, we control the isothermal parameters along the admissible mountain-paths via the theory of nonlinear elliptic systems by E. Heinz.

ADAPTIVE TRACKING CONTROL FOR TWIN ROTOR MULTIPLE-INPUT MULTIPLE-OUTPUT BASED ON ISS STABILIZATION

This paper proposes the angle tracking control method for Twin rotor multi-input multiple-output (TRMS) using the input-to-state stability theory (ISS) for nonlinear systems. To apply this theory, the model of TRMS is rewritten by an Euler-Lagrange forced model with uncertain parameters and input disturbances. The uncertain parameters are the potential energies depended on the mass of TRMS’parts and the input disturbances are considered the friction force, the flat cable force, the effects of the speed of the main rotor on the horizontal movement and the speed of tail rotor to the vertical movements. Using modificated model of TRMS, we design the adaptive controller for angle ISS stabilization to attenuate the influences of uncertain parameters and input disturbances to the angles of TRMS. The robustness of the closed system is shown by the the stabilization of the angles with the yaw and pitch external disturbances, the simulation and experimental results help to proof the rightness of proposed method.

Control problems of Chen–Lee system by adaptive control method

This paper investigates the control problems of Chen–Lee system. Based on control theory of nonlinear systems, the adaptive controllers were proposed to achieve stabilization, synchronization, coexistence of synchronization and anti-synchronization, and projective synchronization of such system. It should be pointed out that the obtained controllers are not only simpler than the existing ones, but also are physical. It is also noted that the obtained methods can be extended to a general chaotic system. Numerical simulations are used to verify the validity and effectiveness of the obtained results.

Thermal Stability Study of Tilted-Polarizer Spin Torque Nano-oscillator

The spin torque nano-oscillator (STNO) with out of film plane fixed layer magnetization, namely the tilted-polarizer spin torque nano-oscillator (TP-STNO), has the advantage of microwave generation without the help of a magnetic field, which makes it promising to be integrated into a chip. In this paper, we present a thermal stability study of TP-STNO by macrospin modeling, in which the thermal noise is induced into the H eff of the Landau-Liftshitz-Gilbert-Slonczewski equation in the form of a stochastic magnetic field. The TP-STNO with different tilted angles is studied by numerical simulations based on the macrospin model and nonlinear system theory. Depending on the analysis of Lyapunov exponents as a function of drive current density in the room temperature range, we obtain the drive current density ranges in which TP-STNO can oscillate stably in the presence of thermal noise. These quantified results are essential during the practical usage of TP-STNO.

Model Reduction of Neutral Linear and Nonlinear Time-Invariant Time-Delay Systems With Discrete and Distributed Delays

The problem of model reduction by moment matching for linear and nonlinear differential time-delay systems is studied. The class of models considered includes neutral differential time-delay systems with discrete-delays and distributed-delays. The description of moment is revisited by means of a Sylvester-like equation for linear time-delay systems and by means of the center manifold theory for nonlinear time-delay systems. In addition the moments at infinity are characterized for both linear and nonlinear time-delay systems. Parameterized families of models achieving moment matching are given. The parameters can be exploited to derive delay-free reduced order models or time-delay reduced order models with additional properties, e.g. , interpolation at an arbitrary large number of points. Finally, the problem of obtaining a reduced order model of an unstable system is discussed and solved.

Dynamics and Stability of Phase Controlled Oscillators

This paper systematically analyzes linear oscillators, e.g., spring-mass-damper systems or RLC-circuits that are controlled by an extension of a phase-locked loop (PLL). These systems are often used in measurement applications where the stability and dynamics directly influence the measurement quality. Therefore, a description of the control loop in terms of phase signals is sought. However, the classical oscillator turns into a highly nonlinear system when it is formulated in amplitude/phase-variables of its input and output signals. Up to now, there were made either ab-initio assumptions of slowly varying parameters or trial-and-error designs. The novel approach proposed in this paper derives a universally valid description in state space form which enables the use of standard methods of nonlinear system theory. Using this description, the stability of phase controlled oscillators is analyzed by means of Lyapunov functions. A linearization is applied in order to effectively design the controller and optimize the closed-loop dynamics. Simulations with the original nonlinear systems are conducted to justify the linear approach. Thereby, two application scenarios are under consideration: Tracking of the desired target value (target phase shift) and resonance tracking (changes of the system parameters). It is found that including the phase dynamics of the oscillator significantly improves the description of the closed-loop behavior. Finally, the results are validated experimentally for an application measuring the viscosity of fluids.

Symmetric and asymmetric period-1 motions in a periodically forced, time-delayed, hardening Duffing oscillator

In this paper, complex symmetric and asymmetric period-1 motions in a periodically forced, time-delayed, hardening Duffing oscillator are predicted through an implicit discrete map. The implicit discrete map is obtained by the discretization of a second-order differential equation of the time-delayed Duffing oscillator. Using the theory of nonlinear discrete systems, period-1 motions in the time-delayed Duffing oscillator are determined analytically from the fixed points of the mapping structures of discrete nodes under the computational accuracy of $$10^{-10}$$ , and the corresponding stability and bifurcation of period-1 motions are determined by eigenvalue analysis. From the discrete nodes of period-1 motions, the finite discrete Fourier series is employed to determine the frequency–amplitude characteristics of period-1 motions in the time-delayed, Duffing oscillator. The stable and unstable, symmetric and asymmetric period-1 motions are presented, and the corresponding stability and bifurcations are illustrated clearly. Presented are the quantity levels of harmonic amplitudes which indicate the accuracy of the semi-analytical solutions of period-1 motions in such a time-delayed oscillator. From the analytical prediction, numerical simulations of complex period-1 motions in the time-delayed, hardening Duffing oscillator are completed. The amplitude spectrums of period-1 motions in the time-delayed Duffing oscillator are given, and the approximate, analytical expressions of period-1 motions can be obtained. For small excitation frequency, discrete time step size should be reduced to keep the same computational accuracy of discrete nodes. This method can be applied to the time-varying time-delayed nonlinear dynamical systems and other nonlinear dynamical systems.

Design of Lyapunov Based Nonlinear Position Control of Electrohydraulic Servo Systems

This paper studies the nonlinear closed loop control of an electrohydraulic servo system. The control strategy is developed based on Lyapunov theory of nonlinear systems using integrator backstepping approach. Highly nonlinear nature of electrohydraulic servo system is well known. Main reasons for nonlinear and non-differentiable mathematical description of systems dynamics are the fluid compressibility, leakage flows, friction forces and nonlinear fluid flow through servo valve orifices. These nonlinear terms influence also the dynamic errors of the control system. Two different nonlinear design procedures are employed feedback linearization and integrator backstepping. Backstepping is used here because it is a powerful and robust nonlinear strategy. These techniques are used for construction of nonlinear control algorithm. The effectiveness of it, to stabilize any operating point of the system is proved by computer simulation. The systems error dynamics significantly depends on tuning parameters of the controller. The wrong selection of these parameters may lead to saturation or chattering of control signals. All derived results are validated by computer simulation of a nonlinear mathematical model of the system. The results are also compared to these obtained with a conventional P controller to prove that classic linear controllers fail to achieve a good tracking of the desired output, especially, when the hydraulic actuator operates at the maximum load. The research studies represented in the paper shows big potential of Lyapunov based nonlinear controller design procedures, to obtain desired control objectives.

Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control.

In this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace spanned by specially chosen observable functions. The Koopman operator is an infinite-dimensional linear operator that evolves functions of the state of a dynamical system. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems. Choosing the right nonlinear observable functions to form an invariant subspace where it is possible to obtain linear reduced-order models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis by leveraging a new algorithm to determine relevant terms in a dynamical system by ℓ1-regularized regression of the data in a nonlinear function space; we also show how this algorithm is related to DMD. Finally, we demonstrate the usefulness of nonlinear observable subspaces in the design of Koopman operator optimal control laws for fully nonlinear systems using techniques from linear optimal control.

Applied Koopman operator theory for power systems technology

Koopman operator is a composition operator defined for a dynamical system described by nonlinear differential or difference equation. Although the original system is nonlinear and evolves on a finite-dimensional state space, the Koopman operator itself is linear but infinite-dimensional (evolves on a function space). This linear operator captures the full information of the dynamics described by the original nonlinear system. In particular, spectral properties of the Koopman operator play a crucial role in analyzing the original system. In the first part of this paper, we review the so-called Koopman operator theory for nonlinear dynamical systems, with emphasis on modal decomposition and computation that are direct to wide applications. Then, in the second part, we present a series of applications of the Koopman operator theory to power systems technology. The applications are established as data-centric methods, namely, how to use massive quantities of data obtained numerically and experimentally, through spectral analysis of the Koopman operator: coherency identification of swings in coupled synchronous generators, precursor diagnostic of instabilities in the coupled swing dynamics, and stability assessment of power systems without any use of mathematical models. Future problems of this research direction are identified in the last concluding part of this paper.

Hopf Bifurcation of Compound Stochastic van der Pol System

Hopf bifurcation analysis for compound stochastic van der Pol system with a bound random parameter and Gaussian white noise is investigated in this paper. By the Karhunen-Loeve (K-L) expansion and the orthogonal polynomial approximation, the equivalent deterministic van der Pol system can be deduced. Based on the bifurcation theory of nonlinear deterministic system, the critical value of bifurcation parameter is obtained and the influence of random strengthδand noise intensityσon stochastic Hopf bifurcation in compound stochastic system is discussed. At last we found that increasedδcan relocate the critical value of bifurcation parameter forward while increasedσmakes it backward and the influence ofδis more sensitive thanσ. The results are verified by numerical simulations.

Robust output voltage control of multimode non-inverting DC–DC converter

ABSTRACTLinear robust controller design based on Uncertainty and Disturbance Estimator theory for nonlinear uncertain single-input single-output systems with external disturbances is discussed in the paper with application to the output voltage control of a unidirectional universal multimode buck/boost/buck–boost converter. The converter is characterized by different nonlinear static and dynamic behaviour in each of the operating modes. The proposed controller forces the system to maintain nearly nominal performance through the whole range of operation space by appropriately estimating and cancelling the terms including nonlinearities, parametric uncertainties and external disturbances. In addition, measurement noise is included in the analysis to demonstrate the trade-offs of the proposed method. Despite the non-triviality and relative complexity of the method, the resulting controller structure is simple, allowing low-cost, low-part-count analogue implementation. Extended simulation results are presented to demonstrate the effectiveness of the proposed control approach.