About the Stabilization of a Nonlinear Perturbed Difference Equation

This paper investigates the local asymptotic stabilization of a very general class of instable autonomous nonlinear difference equations which are subject to perturbed dynamics which can have a different order than that of the nominal difference equation. In the general case, the controller consists of two combined parts, namely, the feedback nominal controller which stabilizes the nominal (i.e., perturbation-free) difference equation plus an incremental controller which completes the stabilization in the presence of perturbed or unmodeled dynamics in the uncontrolled difference equation. A stabilization variant consists of using a single controller to stabilize both the nominal difference equation and also the perturbed one under a small-type characterization of the perturbed dynamics. The study is based on Banach fixed point principle, and it is also valid with slight modification for the stabilization of unstable oscillatory solutions.

Difference equation approach to two-thermocouple sensor characterisation in constant velocity flow environments

Thermocouples are one of the most popular devices for temperature measurement due to their robustness, ease of manufacture and installation, and low cost. However, when used in certain harsh environments, for example, in combustion systems and engine exhausts, large wire diameters are required, and consequently the measurement bandwidth is reduced. This article discusses a software compensation technique to address the loss of high frequency fluctuations based on measurements from two thermocouples. In particular, a difference equation sDEd approach is proposed and compared with existing methods both in simulation and on experimental test rig data with constant flow velocity. It is found that the DE algorithm, combined with the use of generalized total least squares for parameter identification, provides better performance in terms of time constant estimation without any a priori assumption on the time constant ratios of the thermocouples.

Exploiting a priori time constant ratio information in difference equation two-thermocouple sensor characterisation

The characterization of thermocouple sensors for temperature measurement in varying-flow environments is a challenging problem. Recently, the authors introduced novel difference-equation-based algorithms that allow in situ characterization of temperature measurement probes consisting of two-thermocouple sensors with differing time constants. In particular, a linear least squares (LS) lambda formulation of the characterization problem, which yields unbiased estimates when identified using generalized total LS, was introduced. These algorithms assume that time constants do not change during operation and are, therefore, appropriate for temperature measurement in homogenous constant-velocity liquid or gas flows. This paper develops an alternative ß-formulation of the characterization problem that has the major advantage of allowing exploitation of a priori knowledge of the ratio of the sensor time constants, thereby facilitating the implementation of computationally efficient algorithms that are less sensitive to measurement noise. A number of variants of the ß-formulation are developed, and appropriate unbiased estimators are identified. Monte Carlo simulation results are used to support the analysis.

Difference equation sensor characterization algorithms for two-thermocouple probes

The characterization of thermocouple sensors for temperature measurement in variable flow environments is a challenging problem. In this paper, novel difference equation-based algorithms are presented that allow in situ characterization of temperature measurement probes consisting of two-thermocouple sensors with differing time constants. Linear and non-linear least squares formulations of the characterization problem are introduced and compared in terms of their computational complexity, robustness to noise and statistical properties. With the aid of this analysis, least squares optimization procedures that yield unbiased estimates are identified. The main contribution of the paper is the development of a linear two-parameter generalized total least squares formulation of the sensor characterization problem. Monte-Carlo simulation results are used to support the analysis.

Difference equation approach to two-thermocouple sensor characterization in constant velocity flow environments

Thermocouples are one of the most popular devices for temperature measurement due to their robustness, ease of manufacture and installation, and low cost. However, when used in certain harsh environments, for example, in combustion systems and engine exhausts, large wire diameters are required, and consequently the measurement bandwidth is reduced. This article discusses a software compensation technique to address the loss of high frequency fluctuations based on measurements from two thermocouples. In particular, a difference equation (DE) approach is proposed and compared with existing methods both in simulation and on experimental test rig data with constant flow velocity. It is found that the DE algorithm, combined with the use of generalized total least squares for parameter identification, provides better performance in terms of time constant estimation without any a priori assumption on the time constant ratios of the thermocouples.

On a non-autonomous difference equation

Abu-Saris and DeVault proposed two open problems about the difference equation x(n+1) = a(n)x(n)/x(n-1), n = 0, 1, 2,..., where a(n) not equal 0 for n = 0, 1, 2..., x(-1) not equal 0, x(0) not equal 0. In this paper we provide solutions to the two open problems. (c) 2004 Elsevier Inc. All rights reserved.

Analog of Favard's Theorem for Polynomials Connected with Difference Equation of 4-th Order

Orthonormal polynomials on the real line {pn (λ)} n=0 ... ∞ satisfy the recurrent relation of the form: λn−1 pn−1 (λ) + αn pn (λ) + λn pn+1 (λ) = λpn (λ), n = 0, 1, 2, . . . , where λn > 0, αn ∈ R, n = 0, 1, . . . ; λ−1 = p−1 = 0, λ ∈ C. In this paper we study systems of polynomials {pn (λ)} n=0 ... ∞ which satisfy the equation: αn−2 pn−2 (λ) + βn−1 pn−1 (λ) + γn pn (λ) + βn pn+1 (λ) + αn pn+2 (λ) = λ2 pn (λ), n = 0, 1, 2, . . . , where αn > 0, βn ∈ C, γn ∈ R, n = 0, 1, 2, . . ., α−1 = α−2 = β−1 = 0, p−1 = p−2 = 0, p0 (λ) = 1, p1 (λ) = cλ + b, c > 0, b ∈ C, λ ∈ C. It is shown that they are orthonormal on the real and the imaginary axes in the complex plane ...

On Solutions of Holonomic Divided-Difference Equations on Non-Uniform Lattices

The main aim of this paper is the development of suitable bases (replacing the power basis x^n (n\in\IN_\le 0) which enable the direct series representation of orthogonal polynomial systems on non-uniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows to write solutions of arbitrary divided-difference equations in terms of series representations extending results given in [16] for the q-case. Furthermore it enables the representation of the Stieltjes function which can be used to prove the equivalence between the Pearson equation for a given linear functional and the Riccati equation for the formal Stieltjes function. If the Askey-Wilson polynomials are written in terms of this basis, however, the coefficients turn out to be not q-hypergeometric. Therefore, we present a second basis, which shares several relevant properties with the first one. This basis enables to generate the defining representation of the Askey-Wilson polynomials directly from their divided-difference equation. For this purpose the divided-difference equation must be rewritten in terms of suitable divided-difference operators developed in [5], see also [6].

Stability of solutions of linear difference equations with periodic coefficients

This paper represents the last technical contribution of Professor Patrick Parks before his untimely death in February 1995. The remaining authors of the paper, which was subsequently completed, wish to dedicate the article to Patrick. A frequency criterion for the stability of solutions of linear difference equations with periodic coefficients is established. The stability criterion is based on a consideration of the behaviour of a frequency hodograph with respect to the origin of coordinates in the complex plane. The formulation of this criterion does not depend on the order of the difference equation.

Oscillation Criteria of Second-Order Quasi-Linear Neutral Delay Difference Equations

2000 Mathematics Subject Classification: 39A10.

Adaptive instantaneous frequency estimation: Techniques and algorithms

This thesis deals with the problem of the instantaneous frequency (IF) estimation of sinusoidal signals. This topic plays significant role in signal processing and communications. Depending on the type of the signal, two major approaches are considered. For IF estimation of single-tone or digitally-modulated sinusoidal signals (like frequency shift keying signals) the approach of digital phase-locked loops (DPLLs) is considered, and this is Part-I of this thesis. For FM signals the approach of time-frequency analysis is considered, and this is Part-II of the thesis. In part-I we have utilized sinusoidal DPLLs with non-uniform sampling scheme as this type is widely used in communication systems. The digital tanlock loop (DTL) has introduced significant advantages over other existing DPLLs. In the last 10 years many efforts have been made to improve DTL performance. However, this loop and all of its modifications utilizes Hilbert transformer (HT) to produce a signal-independent 90-degree phase-shifted version of the input signal. Hilbert transformer can be realized approximately using a finite impulse response (FIR) digital filter. This realization introduces further complexity in the loop in addition to approximations and frequency limitations on the input signal. We have tried to avoid practical difficulties associated with the conventional tanlock scheme while keeping its advantages. A time-delay is utilized in the tanlock scheme of DTL to produce a signal-dependent phase shift. This gave rise to the time-delay digital tanlock loop (TDTL). Fixed point theorems are used to analyze the behavior of the new loop. As such TDTL combines the two major approaches in DPLLs: the non-linear approach of sinusoidal DPLL based on fixed point analysis, and the linear tanlock approach based on the arctan phase detection. TDTL preserves the main advantages of the DTL despite its reduced structure. An application of TDTL in FSK demodulation is also considered. This idea of replacing HT by a time-delay may be of interest in other signal processing systems. Hence we have analyzed and compared the behaviors of the HT and the time-delay in the presence of additive Gaussian noise. Based on the above analysis, the behavior of the first and second-order TDTLs has been analyzed in additive Gaussian noise. Since DPLLs need time for locking, they are normally not efficient in tracking the continuously changing frequencies of non-stationary signals, i.e. signals with time-varying spectra. Nonstationary signals are of importance in synthetic and real life applications. An example is the frequency-modulated (FM) signals widely used in communication systems. Part-II of this thesis is dedicated for the IF estimation of non-stationary signals. For such signals the classical spectral techniques break down, due to the time-varying nature of their spectra, and more advanced techniques should be utilized. For the purpose of instantaneous frequency estimation of non-stationary signals there are two major approaches: parametric and non-parametric. We chose the non-parametric approach which is based on time-frequency analysis. This approach is computationally less expensive and more effective in dealing with multicomponent signals, which are the main aim of this part of the thesis. A time-frequency distribution (TFD) of a signal is a two-dimensional transformation of the signal to the time-frequency domain. Multicomponent signals can be identified by multiple energy peaks in the time-frequency domain. Many real life and synthetic signals are of multicomponent nature and there is little in the literature concerning IF estimation of such signals. This is why we have concentrated on multicomponent signals in Part-H. An adaptive algorithm for IF estimation using the quadratic time-frequency distributions has been analyzed. A class of time-frequency distributions that are more suitable for this purpose has been proposed. The kernels of this class are time-only or one-dimensional, rather than the time-lag (two-dimensional) kernels. Hence this class has been named as the T -class. If the parameters of these TFDs are properly chosen, they are more efficient than the existing fixed-kernel TFDs in terms of resolution (energy concentration around the IF) and artifacts reduction. The T-distributions has been used in the IF adaptive algorithm and proved to be efficient in tracking rapidly changing frequencies. They also enables direct amplitude estimation for the components of a multicomponent

Blind Characterisation of Sensors with Second-Order Dynamic Response

Compensation for the dynamic response of a temperature sensor usually involves the estimation of its input on the basis of the measured output and model parameters. In the case of temperature measurement, the sensor dynamic response is strongly dependent on the measurement environment and fluid velocity. Estimation of time-varying sensor model parameters therefore requires continuous textit{in situ} identification. This can be achieved by employing two sensors with different dynamic properties, and exploiting structural redundancy to deduce the sensor models from the resulting data streams. Most existing approaches to this problem assume first-order sensor dynamics. In practice, however second-order models are more reflective of the dynamics of real temperature sensors, particularly when they are encased in a protective sheath. As such, this paper presents a novel difference equation approach to solving the blind identification problem for sensors with second-order models. The approach is based on estimating an auxiliary ARX model whose parameters are related to the desired sensor model parameters through a set of coupled non-linear algebraic equations. The ARX model can be estimated using conventional system identification techniques and the non-linear equations can be solved analytically to yield estimates of the sensor models. Simulation results are presented to demonstrate the efficiency of the proposed approach under various input and parameter conditions.

Invariant discretizations of partial differential equations

Un algorithme permettant de discrétiser les équations aux dérivées partielles (EDP) tout en préservant leurs symétries de Lie est élaboré. Ceci est rendu possible grâce à l'utilisation de dérivées partielles discrètes se transformant comme les dérivées partielles continues sous l'action de groupes de Lie locaux. Dans les applications, beaucoup d'EDP sont invariantes sous l'action de transformations ponctuelles de Lie de dimension infinie qui font partie de ce que l'on désigne comme des pseudo-groupes de Lie. Afin d'étendre la méthode de discrétisation préservant les symétries à ces équations, une discrétisation des pseudo-groupes est proposée. Cette discrétisation a pour effet de transformer les symétries ponctuelles en symétries généralisées dans l'espace discret. Des schémas invariants sont ensuite créés pour un certain nombre d'EDP. Dans tous les cas, des tests numériques montrent que les schémas invariants approximent mieux leur équivalent continu que les différences finies standard.