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The numerical technique is applied to the prediction of self-sustained oscillations in a two-axis model of the nonlinear system with sampled data. The sampled-data two-axis LST system model, and its stability equation are analyzed along with the exact solution of the stability equation by numerical-iterative techniques.
A new method, the Hilbert-Huang Transform, has been developed for analyzing nonlinear and nonstationary data. The key part of the method is the Empirical Mode Decomposition with which any complicated data set can be decomposed into a finite and often small number of Intrinsic Mode Functions (IMF). An M is defined as any function having the same numbers of zero-crossing and extrema, and also having symmetric envelopes defined by the local maxima and minima respectively. The IMF also admits well-behaved Hilbert transform. This decomposition method is adaptive, and, therefore, highly efficient. Since the decomposition is based on the local characteristic time scale of the data, it is applicable to nonlinear and nonstationary processes. With the Hilbert transform, the Intrinsic Mode Functions yield instantaneous frequencies'as functions of time that give sharp identifications of imbedded structures. The final presentation of the results is an energy-frequency-time distribution, designated as the Hilbert Spectrum. With this technique we can examine the detailed dynamics characteristics of a nonlinear system through the instantaneous frequency rather than harmonics. Thus it constitutes a new view of the nonlinear dynamics. Examples of classic nonlinear equations and other nonlinear and nonstationary data sets will be used as examples to illustrate the advantage of the application of this new data analysis method.
Advances in theory of modeling errors reported. Recent paper on errors in mathematical models of deterministic linear or weakly nonlinear systems. Extends theoretical work described in NPO-16661 and NPO-16785. Presents concrete way of accounting for difference in structure between mathematical model and physical process or system that it represents.
Systematic attack design is essential to understanding the vulnerabilities of cyber–physical systems (CPSs), to better design for resiliency. In particular, false data injection attacks (FDIAs) are well-known and have been shown to be capable of bypassing bad data detection (BDD) while causing targeted biases in resulting state estimates. However, their effectiveness against moving horizon estimators (MHE) is not well understood. In fact, this paper shows that conventional FDIAs are generally ineffective against MHE. One of the main reasons is that the moving window renders the static FDIA recursively infeasible. Here, this paper proposes a new attack methodology, moving-horizon FDIA (MH-FDIA), by considering both the performance of historical attacks and the current system’s status. Theoretical guarantees for successful attack generation and recursive feasibility are given. Numerical simulations on the IEEE-14 bus system further validate the theoretical claims and show that the proposed MH-FDIA outperforms state-of-the-art counterparts in both stealthiness and effectiveness. In addition, an experiment on a path-tracking control system of an autonomous vehicle shows the feasibility of the MH-FDIA in real-world nonlinear systems.
Several evidences for the directly driven aspect of magnetospheric substorms are presented by reinterpreting what have been thought to be supporting evidences for the unloading process. It is emphasized that some of the confusions in substorm studies could be resolved by understanding that the magnetospheric substorm is primarily a directly driven phenomenon, but has a variety of internal processes. A method is suggested for identification of the directly driven and the unloading components. It is also demonstrated that the magnetosphere is intrinsically a nonlinear system and that a quantitative study of magnetospheric substorms is not possible without taking into account this nonlinearity.
Results are presented that demonstrate how good reduced-order models can be obtained directly by recursive parameter identification using input/output (I/O) data of high-order nonlinear systems. Three different methods of obtaining a third-order linear model from a seventh-order nonlinear turbojet engine model are compared. The first method is to obtain a linear model from the original model and then reduce the linear model by standard reduction techniques such as residualization and balancing. The second method is to identify directly a third-order linear model by recursive least-squares parameter estimation using I/O data of the original model. The third method is to obtain a reduced-order model from the original model and then linearize the reduced model. Frequency responses are used as the performance measure to evaluate the reduced models. The reduced-order models along with their Bode plots are presented for comparison purposes.
Certification of adaptive control technologies for both manned and unmanned aircraft represent a major challenge for current Verification and Validation techniques. A (missing) key step towards flight certification of adaptive flight control systems is the definition and development of analysis tools and methods to support Verification and Validation for nonlinear systems, similar to the procedures currently used for linear systems. In this paper, we describe and demonstrate the advantages of L(sub l) adaptive control architectures for closing some of the gaps in certification of adaptive flight control systems, which may facilitate the transition of adaptive control into military and commercial aerospace applications. As illustrative examples, we present the results of a piloted simulation evaluation on the NASA AirSTAR flight test vehicle, and results of an extensive flight test program conducted by the Naval Postgraduate School to demonstrate the advantages of L(sub l) adaptive control as a verifiable robust adaptive flight control system.
A method is proposed for dealing with the divergence phenomenon problem encountered in applications of minimum-variance recursive estimators when the error covariance calculated by the estimator becomes inconsistent with the actual error covariance. The proposed method differs from previous ones in that the state model and statistics are accepted as given. The form of the optimal estimator is used, but a constant check is made on the consistency of the calculated and actual error covariances. The method is independent of the source of error, whether it be inaccuracies in the system model, incorrect values of the a priori and random process statistics, approximations required in the case of nonlinear systems, or computational roundoff. Several simulated examples, in which inconsistencies in the calculated and actual error covariances exist, show a significant improvement in the performance of the estimator when the given procedure is applied.
A phenomena is described occuring in wave propagation for systems of conservation laws which are not hyperbolic. The nature of the wave propagation for linear systems of conservation laws which are degenerate in the sense that they are not equivalent to diagonal systems is examined. It is concluded that in the degenerate case, 'resonance' occurs, that is, the solutions are combinations of traveling waves and 'resonance waves'. Here resonance means that the solution may become unbounded even if the initial values are bounded. The solutions (waves) can be described as a superposition of 'packets' (groups) of 'resonance waves' traveling with the same speed. These are linear systems with constant coefficients which are self-exciting, which is what one would expect to encounter in nonlinear systems.
In this work, we implement a tensor network-based quantum algorithm to solve unsteady, nonlinear partial differential equations (PDEs). The challenge lies in how to effectively represent, encode, process, and evolve the nonlinear system of PDEs on quantum computers. We will discuss the new techniques using the compressible 1-dimensional (1D) Burgers' equation as an example, because it represents the fundamental nonlinear feature and yet removes certain complexity in physics, allowing us to focus on the design of quantum algorithms. Previous attempts to solve nonlinear PDEs in quantum computation have often involved storing multiple copies of solutions or employing linearizations. Neither is practical due to exponential scaling with evolution time or insufficient solution accuracy. Our framework is based on matrix product states (MPSs) and matrix product operators (MPOs). For example, the velocity field is represented by MPS, whereas the linear and nonlinear spatial differential terms of the velocity field are processed by MPOs. Our primary focus herein is to verify and validate the various tensor network components of the algorithm using solutions obtained by the classical algorithms on high performance computing (HPC) architectures. We use a classical time marching method to demonstrate the functionality of the tensor network operations to model the PDE and their robustness with the time evolution of the system. Our classical simulation results demonstrate the utility of tensor network-based operations in modeling nonlinear PDEs and highlight the necessity as well as potential advantages of using quantum simulations for these techniques.
We observe that the so-called Generalised Equipartition Law for hamiltonian systems is actually valid only under specific hypotheses – unfortunately omitted in some textbooks – which limit its applicability when dealing with nonlinear systems. We introduce a new coordinate-independent generalisation which overcomes this problem and can be applied to a larger set of functions. A simple example of application is discussed.
This paper presents a dynamic study of the Wildwood Pendulum, a commercially available desktop system which exhibits a strange attractor. The purpose of studying this chaotic pendulum is twofold: to gain insight in the paradigmatic approach of modeling, simulating, and determining chaos in nonlinear systems; and to provide a desktop model of chaos as a visual tool. For this study, the nonlinear behavior of this chaotic pendulum is modeled, a computer simulation is performed, and an experimental performance is measured. An assessment of the pendulum in the phase plane shows the strange attractor. Through the use of a box-assisted correlation dimension methodology, the attractor dimension is determined for both the model and the experimental pendulum systems. Correlation dimension results indicate that the pendulum and the model are chaotic and their fractal dimensions are similar.
Oscillatory properties of linear matrix differential system, applying generalization of polar coordinate transformation to certain systems of second order
It has become more and more evident that many difficulties encountered in the course of aircraft flutter analyses can be traced to strong localized nonlinearities in the control mechanisms. To cope with these problems, more reliable mathematical models paying special attention to control system nonlinearities were established by means of modified ground vibration test procedures in combination with suitably adapted modal synthesis approaches. Three different concepts are presented.
Report discusses global instability in orbital motion of two-link planar mechanism. Principal objective, contributes to understanding of chaotic motions in robot manipulators and other deterministic mechanical systems. Discussion begins with brief review of previous studies of chaotic motion and introduces notion of orbital instability in nonlinear systems. Introduces geometric approach useful in representation of orbital instability.
The hunt (self-oscillations) of a physical system may often be removed by the introduction of an appropriate stabilizing signal which changed the open loop gain in a nonlinear manner. More generally, the performance of nonlinear systems in many cases may be improved by the introduction of extra signals. The theory of signal stabilization developed here extends the earlier work by Oldenburger and Liu involving an equivalent gain concept. It is shown that with the aid of the Fourier series the designer can determine the periodic signal to be inserted at one point in a loop to yield a desired stabilizing input to a nonlinear element in the loop. The use of sinusoidal and triangular inputs to a limiter are compared. An example where a limiter is the only nonlinearity is employed to illustrate the theory. The approach developed here explains experimental results previously reported by Oldenburger.
Existence theorem proved for nonlinear systems. Report discusses mathematical requirements for existence of solution to minimum-time control problem for robotic manipulator that obeys nonlinear equations of motion.
Control of the X33 reusable launch vehicle is considered. The launch control problem consists of automatic tracking of the launch trajectory which is assumed to be optimally precalculated. It requires development of a reliable, robust control algorithm that can automatically adjust to some changes in mission specifications (mass of payload, target orbit) and the operating environment (atmospheric perturbations, interconnection perturbations from the other subsystems of the vehicle, thrust deficiencies, failure scenarios). One of the effective control strategies successfully applied in nonlinear systems is the Sliding Mode Control. The main advantage of the Sliding Mode Control is that the system's state response in the sliding surface remains insensitive to certain parameter variations, nonlinearities and disturbances. Employing the time scaling concept, a new two (three)-loop structure of the control system for the X33 launch vehicle was developed. Smoothed sliding mode controllers were designed to robustly enforce the given closed-loop dynamics. Simulations of the 3-DOF model of the X33 launch vehicle with the table-look-up models for Euler angle reference profiles and disturbance torque profiles showed a very accurate, robust tracking performance.