Orthonormal basis functions for modelling continuous-time systems

In this paper we address the identification problem of a continuous-time deterministic system. We consider that the continuous-time system can be approximated by using the orthonormal continuous basis function of Laguerre. We assume that only discrete-time measurements are available and we obtain the exact discrete-time model in terms of the continuous-time model parameters of Laguerre basis function, the sample time and the Euler-Fröbenius polynomials. The estimation problem is solved by using the Least Squares method. We illustrate the benefits of our proposal with numerical simulations.

In this paper, model sets for continuous-time linear time invariant systems that are spanned by fixed pole orthonormal bases are investigated. These bases generalise the well known Laguerre and Kautz bases. It is shown that the obtained model sets are complete in all of the Hardy spaces Hp(Π), 1 ≤ p p (Π), 1 ≤ p p (Π), 1 ≤ p < ∞. The results in this paper have application in system identification, model reduction and control system synthesis.

In many engineering applications, continuous-time models are preferred to discrete-time ones, in that they provide good physical insight and can be derived also from non-uniformly sampled data. However, for such models, model selection is a hard task if no prior physical knowledge is given. In this paper, we propose a non-parametric approach to infer a continuous-time linear model from data, by automatically selecting a proper structure of the transfer function and guaranteeing to preserve the system stability properties. By means of benchmark simulation examples, the proposed approach is shown to outperform state-of-the-art continuous-time methods, also in the critical case when short sequences of canonical input signals, like impulses or steps, are used for model learning.

A procedure for identifying linear continuous-time multivariable systems is presented. The scheme is based on a discrete identification method which requires no structural identification. This is accomplished by the use of pseudo-observable forms whose structure is defined by their corresponding pseudo-observability indices. A First-Order Hold transformation of this discrete-time model to an equivalent continuous-time model is then performed. First, the state transition matrix is transformed to an equivalent continuous-time system matrix using a procedure which is insensitive to the size of the spectral radii of the system. The remaining system matrices are then transformed using a generalized First-Order Hold transformation technique which places no restrictions on the singularity of the continuous-time system.

In hands-free telephone or video conference application, there exists an acoustic feedback coupling between the loudspeaker and microphone in an enclosed environment, which creates the acoustic echo. FIR filters are commonly used in acoustic echo cancellers because of their simple structure. However, in this paper, the Kautz and Laguerre filter structures are shown to be more efficient echo cancellers than the FIR filters, because they can describe accurately the acoustic echo system with fewer parameters. These filters are built from their respective orthonormal Kautz and Laguerre basis functions. The proposal is motivated by some theoretical and numerical results that the time-varying acoustic echo path is basically due to its time-varying zeros and not its time-invariant acoustical poles. Therefore, the poles of the Kautz and the Laguerre filters are estimated, and can be kept fixed or updated occasionally if required. The poles are estimated by a batch Gauss-Newton algorithm. Then, the coefficients of the Kautz and Laguerre filters can be estimated by most recursive algorithms that are suitable for linear regression models, e.g., the normalized LMS algorithm. Generally, it is shown that the proposed Kautz and Laguerre filters, as the filter structures in an acoustic echo canceller, have better convergence and tracking properties than the FIR and IIR filters.

Presents the analyses of the stability and robustness of multivariable continuous-time nonlinear systems subject to parameter uncertainties and with digital fuzzy controllers. To proceed with the analysis, first, an uncertain multivariable nonlinear plant is represented by a fuzzy plant model with parameter uncertainties. Second, a digital fuzzy controller is designed to close the feedback loop. Third, the stability criteria, the robust area and the largest sampling period are derived in terms of the matrix measures of the system parameters and the norms of the parameter uncertainties. An application example on stabilizing an uncertain nonlinear mass-spring-damper system is given to show the stabilizability and robustness properties of the proposed digital fuzzy controller.

Identification of continuous-time nonlinear systems: The nonlinear difference equation with moving average noise (NDEMA) framework

Identification with Generalized Orthonormal Basis Functions - Statistical Analysis and Error Bounds

This paper studies synchronization in an array of coupled neural networks with Markovian jumping and random coupling strength. The array of neural networks are coupled in a random fashion which is governed by Bernoulli random variable and each node has an interval time-varying delay. By designing a novel Lyapunov functional, using some inequalities and the properties of random variables, several delay-dependent synchronization criteria are derived for the coupled networks of continuous-time version. Discrete-time analogues of the continuous-time networks are also formulated and studied. Some new lemmas are developed to obtain less conservative synchronization criteria of both continuous-time model and its discrete-time analogues. Numerical examples of both continuous-time system and its discrete-time analogues are finally given to demonstrate the effectiveness of the theoretical results.

Continuous-Time System Monitoring via a Continuous-Time Model Parameter Estimation

In the modeling of continuous time systems, there is a need to convert the analog system to a discrete time system. The transform frequently encountered in the literature is the impulse invariant transform. This transform has proven to be inadequate for biological system modeling. In this paper, we review the available linear transforms and derive three new transforms, namely the centered step invariant transform, the local cubic spline invariant transform, and the scaled impulse invariant transform.

A Sentiment-Based Model for the Bitcoin: Theory, Estimation and Option Pricing

Presents a framework for fuzzy modeling of dynamic systems using orthonormal basis functions in the representation of the model input signals. The main objective of using orthonormal bases is to overcome the task of estimating the order and time delay of the process. The result is a nonlinear moving average fuzzy model which, consequently, has no feedback of prediction errors. Although any technique of fuzzy modeling can be used in the proposed framework, a relational approach is considered. The performance of fuzzy models with orthonormal basis functions is illustrated by examples and the results are compared with those provided by conventional fuzzy models and Volterra models.

A fuzzy clustering approach is developed to select pole locations for Orthonormal Basis Functions (OBFs), used for identification of Linear Parameter Varying (LPV) systems. The identification approach is based on interpolation of locally identified Linear Time Invariant (LTI) models, using globally fixed OBFs. Selection of the optimal OBF structure, that guarantees the least worst-case local modelling error in an asymptotic sense, is accomplished through the fusion of the Kolmogorov n-width (KnW) theory and Fuzzy c-Means (FcM) clustering of observed sample system poles.

An alternative method was proposed for identification of linear continuous time system with noisy output. Using Gaussian modulating filters, by numerical integration, a discrete equivalent identification model that was parametrized directly in continuous time model parameters was obtained, and the LS estimation algorithm was developed. The LS estimator is biased even with white measurement output noise, and the bias is determined by variance of output noise. On the basis of obtained output measurement noise variance, the bias was compensated and the consistent estimates of system parameters were given. Simulation which compares the standard LS with BCLS algorithm indicates that the proposed algorithm is an efficient method for noisy output continuous time system identification.