Stability radius and internal versus external stability in Banach spaces: an evolution semigroup approach

In this paper the theory of evolution semigroups is developed and used to provide a framework to study the stability of general linear control systems. These include time-varying systems modeled with unbounded state-space operators acting on Banach spaces. This approach allows one to apply the classical theory of strongly continuous semigroups to time-varying systems. In particular, the complex stability radius may be expressed explicitly in terms of the generator of a (evolution) semigroup. Examples are given to show that classical formulas for the stability radius of an autonomous Hilbert-space system fail in more general settings. Upper and lower bounds on the stability radius are provided for these general systems. In addition, it is shown that the theory of evolution semigroups allows for a straightforward operator-theoretic analysis of internal stability as determined by classical frequency-domain and input-output operators, even for nonautonomous Banach-space systemsComment: Also at http://www.math.missouri.edu/~stephen/preprint

Time-Frequency multipliers for sound synthesis

International audienceTime-frequency analysis and wavelet analysis are generally used for providing signal expansions that are suitable for various further tasks such as signal analysis, de-noising, compression, source separation, ... However, time-frequency analysis and wavelet analysis also provide efficient ways for constructing signals' transformations. They are modelled as linear operators that can be designed directly in the transformed domain, i.e. the time-frequency plane, or the time-scale half plane. Among these linear operators, transformations that are diagonal in the time-frequency or time scale spaces, i.e. that may be expressed by multiplications in these domains, deserve particular attention, as they are extremely simple to implement, even though their properties are not necessarily easy to control. This work is a first attempt for exploring such approaches in the context of the analysis and the design of sound signals. We study more specifically the transformations that may be interpreted as linear time-varying (LTV) systems (often called time-varying filters). It is known that under certain assumptions, the latter may be conveniently represented by pointwise multiplication with a certain time frequency transfer function in the time-frequency domain. The purpose of this work is to examine such representations in practical situations, and investigate generalizations. The originality of this approach for sound synthesis lies in the design of practical operators that can be optimized to morph a given sound into another one, at a very high sound quality

Human operator identification model and related computer programs

Four computer programs which provide computational assistance in the analysis of man/machine systems are reported. The programs are: (1) Modified Transfer Function Program (TF); (2) Time Varying Response Program (TVSR); (3) Optimal Simulation Program (TVOPT); and (4) Linear Identification Program (SCIDNT). The TV program converts the time domain state variable system representative to frequency domain transfer function system representation. The TVSR program computes time histories of the input/output responses of the human operator model. The TVOPT program is an optimal simulation program and is similar to TVSR in that it produces time histories of system states associated with an operator in the loop system. The differences between the two programs are presented. The SCIDNT program is an open loop identification code which operates on the simulated data from TVOPT (or TVSR) or real operator data from motion simulators

Applications of statistics in the spectral analysis of time-varying systems

Recent advances in the theory of evolutionary spectral analysis of time-varying systems has led to a resurgence in the popularity of frequency domain analysis techniques. Policies for adaptive control of time-varying systems based on state-space and Liaponov techniques require an accurate measurement of the system phase variables. Under inherently noisy conditions, access to the complete system state is seldom possible, and frequency domain analysis requirinq only input/output measurements has an obvious appeal. The sampling properties of short-term spectral estimates are of central importance both in system tracking and in choosing suitable control policies. Goodman (1957) developed some of the sampling properties associated with spectral estimates of complex bivariate Gaussian processes. Akaike (1962-66) extended Goodman's results to multi input/output linear systems with 'Gaussian input forcing functions. Both these authors considered the case where the data sequences were stationary. This thesis reviews and extends the research of these two authors with respect to single input/output linear systems. It is shown that the sampling distributions associated with spectral estimates of stationary open-loop systems are approximately valid for a restricted class of non-stationary systems. Two examples of non-stationary systems are investigated and an adaptive control technique using input compensation in the frequency domain is developed on a hydraulic fatigue loading rig. It is shown that statistical tests developed earlier can successfully identify system variations when estimates are measured in a noisy environment. The sampling distributions associated with spectral estimates of closed-loop systems are developed and the results are applied to the modelling and tracking of the human operator response in a trackinq task situation, for various input signals. With regard to future research, it remains to extend the results for closed-loop systems to the time-varying multi input/output case. In its full complexity this problem remains intractable but by considering uncorrelated Gaussian inputs it reduces to determining the distributions associated with multi-variate complex Gaussian sequences

Frequency-Domain Data-Driven Controller Synthesis for Unstable LPV Systems

Synthesizing controllers directly from frequency-domain measurement data is a powerful tool in the linear time-invariant framework. Ever-increasing performance requirements necessitate extending these approaches to account for plant variations. The aim of this paper is to develop frequency-domain analysis and synthesis conditions for local internal stability and $\mathcal{H}_\infty$-performance of single-input single-output linear parameter-varying systems. The developed synthesis procedure only requires frequency-domain measurement data of the system and does not need a parametric model of the plant. The capabilities of the synthesis procedure are demonstrated on an unstable nonlinear system.Comment: 8 pages, 8 figures, submitted in Proceedings of the 4th IFAC workshop on Linear Parameter-Varying Systems, 202

Bayesian Non-linear System Identification and Frequency Response Analysis with Application to Soft Smart Actuators

Newly emerging classes of next generation soft-smart actuators are set to have a huge impact on the fields of robotics, orthotics and prosthetics due to their lightweight, high-strain and muscle-like properties. Like muscle, these actuators can be used in multiple roles, e.g. both as actuators and brakes, due their variable compliance. One important class of soft actuator is the dielectric elastomer actuator (DEA). However, DEAs are extremely difficult to control due to their non-linear and time varying dynamics. A crucial step in the advancement of this technology is the development of techniques for systems level modelling and analysis, which is the focus of this thesis. In the first part of the thesis, a set of DEAs are identified and analysed using standard methods from the field of system identification, obtaining non-linear autoregressive with exogenous input (NARX) models. These provide a benchmark against which later methods are evaluated. The key novelty in this part is the development of NARX models of DEAs for use in non-linear frequency-domain analysis. This result provides insight for the first time into how a set of similarly fabricated DEAs vary in different ways. A further aspect of DEA behaviour is their unexplained time varying behaviour. The system identification approach used to identify NARX models of DEAs is in a convenient form such that it can be easily extended to cater for this time varying behaviour. There are however very few available methods for the frequency domain analysis of time varying systems. A novel method for time varying frequency domain analysis of NARX systems is developed in this work and applied to the DEAs. The analysis procedure is used to provide insight on how the dynamic behaviour of DEAs change over time. In the second part of the thesis a novel approach to the joint structure detection and parameter estimation of NARX models is developed using a sparse Bayesian method. The Bayesian framework allows for the estimation of posterior distributions over model parameters, characterising the model uncertainty. Analytic solutions are found that describe model uncertainty in the frequency-domain as confidence bounds on both linear and higher order frequency response functions. The sparse Bayesian identification algorithm is applied to the DEA data sets and is used to give the first non-linear dynamic model of DEAs with uncertainty bounds plus the first description of DEA dynamics in the frequency-domain, again with uncertainty bounds

Modelling and Simulation of Quasi-Resonant Inverter for Induction Heating under Variable Load

Single-switch quasi-resonant DC inverters are preferred in low-power induction-heating applications for their cheapness. However, they pose difficulties in enforcing soft-switching and show limited controllability. A good design of these converters must proceed in parallel with the characterization of the load and the operating conditions. The control of the switching frequency has a critical relationship to the non-linear behavior of the load due to electro-thermal coupling and geometrical anisotropies. Finite element methods enable the analysis of this kind of multiphysics coupled systems, but the simulation of transient dynamics is computationally expensive. The goal of this article is to propose a time-domain simulation strategy to analyze the behavior of induction heating systems with a quasi-resonant single-ended DC inverter using pulse frequency modulation and variable load. The load behavior is estimated through frequency stationary analysis and integrated into the time-domain simulations as a non-linear equivalent impedance parametrized by look-up tables. The model considers variations in temperature dynamics, the presence of work-piece anisotropies, and current harmonic waveforms. The power regulation strategy based on the control of the switch turn-on time is tested in a case study with varying load and it is shown that it is able to maintain the converter in the safe operation region, handling variations up to of (Formula presented.) in the equivalent load resistance

The theory and design of recombination nonuniform filter-banks with linear-phase analysis/synthesis filters

The 47th Midwest Symposium on Circuits and Systems Conference, Salt Lake City, Utah, USA, 25-28 July 2004This paper studies the theory and design of a class of linear-phase (LP) nonuniform filter-banks (FBs) called recombination nonuniform FBs (RNFBs). It is based on a recombination structure, where certain channels of an M-channel uniform FB are merged by synthesis filters of transmultiplexor (TMUX). It is assumed that the numbers of channels of the FB and TMUX are coprime to each other so that it is possible to obtain linear-time invariant (LTI) analysis/synthesis filters, instead of linear periodic time varying (LPTV) filters. The spectral supports of the analysis filters are analyzed, and the existence and matching conditions to obtain LP RNFBs with good frequency characteristics are then derived. The LTI representation of the analysis filters and the use of cosine-roll-off characteristics allow us to design the analysis filters by the REMEZ exchange algorithm. Design examples of LP nearly perfect reconstruction (NPR) RNFBs are given to demonstrate the effectiveness of the proposed method.published_or_final_versio

Electro-optomechanical equivalent circuits for quantum transduction

Using the techniques of optomechanics, a high-$Q$ mechanical oscillator may serve as a link between electromagnetic modes of vastly different frequencies. This approach has successfully been exploited for the frequency conversion of classical signals and has the potential of performing quantum state transfer between superconducting circuitry and a traveling optical signal. Such transducers are often operated in a linear regime, where the hybrid system can be described using linear response theory based on the Heisenberg-Langevin equations. While mathematically straightforward to solve, this approach yields little intuition about the dynamics of the hybrid system to aid the optimization of the transducer. As an analysis and design tool for such electro-optomechanical transducers, we introduce an equivalent circuit formalism, where the entire transducer is represented by an electrical circuit. Thereby we integrate the transduction functionality of optomechanical systems into the toolbox of electrical engineering allowing the use of its well-established design techniques. This unifying impedance description can be applied both for static (DC) and harmonically varying (AC) drive fields, accommodates arbitrary linear circuits, and is not restricted to the resolved-sideband regime. Furthermore, by establishing the quantized input-output formalism for the equivalent circuit, we obtain the scattering matrix for linear transducers using circuit analysis, and thereby have a complete quantum mechanical characterization of the transducer. Hence, this mapping of the entire transducer to the language of electrical engineering both sheds light on how the transducer performs and can at the same time be used to optimize its performance by aiding the design of a suitable electrical circuit.Comment: 30 pages, 9 figure