Estimation of generalised frequency response functions

Volterra series theory has a wide application in the representation, analysis, design and control of nonlinear systems. A new method of estimating the Volterra kernels in the frequency domain is introduced based on a non-parametric algorithm. Unlike the traditional non-parametric methods using the DFT transformed input-output data, this new approach uses the time domain measurements directly to estimate the frequency domain response functions

Super cavity solitons and the coexistence of multiple nonlinear states in a tristable passive Kerr resonator

Passive Kerr cavities driven by coherent laser fields display a rich landscape of nonlinear physics, including bistability, pattern formation, and localised dissipative structures (solitons). Their conceptual simplicity has for several decades offered an unprecedented window into nonlinear cavity dynamics, providing insights into numerous systems and applications ranging from all-optical memory devices to microresonator frequency combs. Yet despite the decades of study, a recent theoretical study has surprisingly alluded to an entirely new and unexplored paradigm in the regime where nonlinearly tilted cavity resonances overlap with one another [T. Hansson and S. Wabnitz, J. Opt. Soc. Am. B 32, 1259 (2015)]. We have used synchronously driven fiber ring resonators to experimentally access this regime, and observed the rise of new nonlinear dissipative states. Specifically, we have observed, for the first time to the best of our knowledge, the stable coexistence of dissipative (cavity) solitons and extended modulation instability (Turing) patterns, and performed real time measurements that unveil the dynamics of the ensuing nonlinear structures. When operating in the regime of continuous wave tristability, we have further observed the coexistence of two distinct cavity soliton states, one of which can be identified as a "super" cavity soliton as predicted by Hansson and Wabnitz. Our experimental findings are in excellent agreement with theoretical analyses and numerical simulations of the infinite-dimensional Ikeda map that governs the cavity dynamics. The results from our work reveal that experimental systems can support complex combinations of distinct nonlinear states, and they could have practical implications to future microresonator-based frequency comb sources.Comment: 13 pages, 6 figure

Can Computer Algebra be Liberated from its Algebraic Yoke ?

So far, the scope of computer algebra has been needlessly restricted to exact algebraic methods. Its possible extension to approximate analytical methods is discussed. The entangled roles of functional analysis and symbolic programming, especially the functional and transformational paradigms, are put forward. In the future, algebraic algorithms could constitute the core of extended symbolic manipulation systems including primitives for symbolic approximations.Comment: 8 pages, 2-column presentation, 2 figure

emgr - The Empirical Gramian Framework

System Gramian matrices are a well-known encoding for properties of input-output systems such as controllability, observability or minimality. These so-called system Gramians were developed in linear system theory for applications such as model order reduction of control systems. Empirical Gramian are an extension to the system Gramians for parametric and nonlinear systems as well as a data-driven method of computation. The empirical Gramian framework - emgr - implements the empirical Gramians in a uniform and configurable manner, with applications such as Gramian-based (nonlinear) model reduction, decentralized control, sensitivity analysis, parameter identification and combined state and parameter reduction

A review of convex approaches for control, observation and safety of linear parameter varying and Takagi-Sugeno systems

This paper provides a review about the concept of convex systems based on Takagi-Sugeno, linear parameter varying (LPV) and quasi-LPV modeling. These paradigms are capable of hiding the nonlinearities by means of an equivalent description which uses a set of linear models interpolated by appropriately defined weighing functions. Convex systems have become very popular since they allow applying extended linear techniques based on linear matrix inequalities (LMIs) to complex nonlinear systems. This survey aims at providing the reader with a significant overview of the existing LMI-based techniques for convex systems in the fields of control, observation and safety. Firstly, a detailed review of stability, feedback, tracking and model predictive control (MPC) convex controllers is considered. Secondly, the problem of state estimation is addressed through the design of proportional, proportional-integral, unknown input and descriptor observers. Finally, safety of convex systems is discussed by describing popular techniques for fault diagnosis and fault tolerant control (FTC).Peer ReviewedPostprint (published version

Towards a Lagrange-Newton approach for PDE constrained shape optimization

The novel Riemannian view on shape optimization developed in [Schulz, FoCM, 2014] is extended to a Lagrange-Newton approach for PDE constrained shape optimization problems. The extension is based on optimization on Riemannian vector space bundles and exemplified for a simple numerical example.Comment: 16 pages, 4 figures, 1 tabl

Multidimensional semi-gap solitons in a periodic potential

The existence, stability and other dynamical properties of a new type of multi-dimensional (2D or 3D) solitons supported by a transverse low-dimensional (1D or 2D, respectively) periodic potential in the nonlinear Schr\"{o}dinger equation with the self-defocusing cubic nonlinearity are studied. The equation describes propagation of light in a medium with normal group-velocity dispersion (GVD). Strictly speaking, solitons cannot exist in the model, as its spectrum does not support a true bandgap. Nevertheless, the variational approximation (VA) and numerical computations reveal stable solutions that seem as completely localized ones, an explanation to which is given. The solutions are of the gap-soliton type in the transverse direction(s), in which the periodic potential acts in combination with the diffraction and self-defocusing nonlinearity. Simultaneously, in the longitudinal (temporal) direction these are ordinary solitons, supported by the balance of the normal GVD and defocusing nonlinearity. Stability of the solitons is predicted by the VA, and corroborated by direct simulations.Comment: European Physical Joournal D, in pres

Techniques for designing rotorcraft control systems

This report summarizes the work that was done on the project from 1 Apr. 1992 to 31 Mar. 1993. The main goal of this research is to develop a practical tool for rotorcraft control system design based on interactive optimization tools (CONSOL-OPTCAD) and classical rotorcraft design considerations (ADOCS). This approach enables the designer to combine engineering intuition and experience with parametric optimization. The combination should make it possible to produce a better design faster than would be possible using either pure optimization or pure intuition and experience. We emphasize that the goal of this project is not to develop an algorithm. It is to develop a tool. We want to keep the human designer in the design process to take advantage of his or her experience and creativity. The role of the computer is to perform the calculation necessary to improve and to display the performance of the nominal design. Briefly, during the first year we have connected CONSOL-OPTCAD, an existing software package for optimizing parameters with respect to multiple performance criteria, to a simplified nonlinear simulation of the UH-60 rotorcraft. We have also created mathematical approximations to the Mil-specs for rotorcraft handling qualities and input them into CONSOL-OPTCAD. Finally, we have developed the additional software necessary to use CONSOL-OPTCAD for the design of rotorcraft controllers