Generalized Orthonormal Basis Functions Research Articles

Continuous-Time Subspace Identification with Prior Information Using Generalized Orthonormal Basis Functions

This paper presents a continuous-time subspace identification method utilizing prior information and generalized orthonormal basis functions. A generalized orthonormal basis is constructed by a rational inner function, and the transformed noises have ergodic properties. The lifting approach and the Hambo system transform are used to establish the equivalent nature of continuous and transformed discrete-time stochastic systems. The constrained least squares method is adopted to investigate the incorporation of prior knowledge in order to further increase the subspace identification algorithm’s accuracy. The input–output algebraic equation derives an optimal multistep forward predictor, and prior knowledge is expressed as equality constraints. In order to solve an optimization problem with equality constraints characterizing the prior knowledge, the proposed method reduces the computational burden. The effectiveness of the proposed method is provided by numerical simulations.

Convolutional Neural Networks as Asymmetric Volterra Models Based on Generalized Orthonormal Basis Functions.

This paper introduces a convolutional neural network (CNN) approach to derive Volterra models of dynamical systems based on generalized orthonormal basis function (GOBF)-Volterra. The approach derives the parameters of the model through a CNN and the neural network's learned weights represent the poles of a system. Simulation results show that the parameters of the system can be exactly recovered when no noise is applied. Furthermore, when noise is present, the errors in the parameters are very small for both the linear and nonlinear cases. Finally, the approach is used to identify the model of a quadcopter using data from actual flight tests. Comparisons with previous works demonstrate that CNNs can be satisfactorily used for the identification of dynamical systems.

The quest for the right kernel in Bayesian impulse response identification: The use of OBFs

Kernel-based regularization approaches for impulse response estimation of Linear Time-Invariant (LTI) systems have received a lot of attention recently. The reason is that regularized least-squares estimators may achieve a favorable bias/variance trade-off compared with classical Prediction Error Minimization (PEM) methods. To fully exploit this property, the kernel function needs to capture relevant aspects of the data-generating system at hand. Hence, it is important to design automatic procedures for kernel design based on data or prior knowledge. The kernel models, so far introduced, focus on encoding smoothness and BIBO-stability of the expected impulse response while other properties, like oscillatory behavior or the presence of fast and slow poles, have not been successfully implemented in kernel design. Inspired by the representation theory of dynamical systems, we show how to build stable kernels that are able to capture particular aspects of system dynamics via the use of Orthonormal Basis Functions (OBFs). In particular, desired dynamic properties can be easily encoded via the generating poles of OBFs. Such poles are seen as hyperparameters which are tuned via marginal likelihood optimization. Special cases of our kernel construction include Laguerre, Kautz, and Generalized OBFs (GOBFs)-based kernel structures. Monte-Carlo simulations show that the OBFs-based kernels perform well compared with stable spline/TC kernels, especially for slow systems with dominant poles close to the unit circle. Moreover, the capability of Kautz basis to model resonating systems is also shown.

Asymmetric Volterra Models Based on Ladder-Structured Generalized Orthonormal Basis Functions

In this paper, an improved method to construct and estimate Volterra models using Generalized Orthonormal Basis Functions (GOBF) is presented. The proposed method extends results obtained in previous works, where an exact technique for optimizing the GOBF parameters (poles) for symmetric Volterra models of any order was presented. The proposed extensions take place in two different ways: (i) the new formulation is derived in such a way that each multidimensional kernel of the model is decomposed into a set of independent orthonormal bases (rather than a single, common basis), each of which is parameterized by an individual set of poles responsible for representing the dominant dynamic of the kernel along a particular dimension; and (ii) the new formulation is based on a ladder-structured GOBF architecture that is characterized by having only real-valued parameters to be estimated, regardless of whether the GOBF poles encoded by these parameters are real- or complex-valued. The exact gradients of an error functional with respect to the parameters to be optimized are computed analytically and provide exact search directions for an optimization process that uses only input-output data measured from the dynamic system to be modeled. Computational experiments are presented to illustrate the benefits of the proposed approach when modeling nonlinear systems.

Wiener system identification with generalized orthonormal basis functions

Many nonlinear systems can be described by a Wiener–Schetzen model. In this model, the linear dynamics are formulated in terms of orthonormal basis functions (OBFs). The nonlinearity is modeled by a multivariate polynomial. In general, an infinite number of OBFs are needed for an exact representation of the system. This paper considers the approximation of a Wiener system with finite-order infinite impulse response dynamics and a polynomial nonlinearity. We propose to use a limited number of generalized OBFs (GOBFs). The pole locations, needed to construct the GOBFs, are estimated via the best linear approximation of the system. The coefficients of the multivariate polynomial are determined with a linear regression. This paper provides a convergence analysis for the proposed identification scheme. It is shown that the estimated output converges in probability to the exact output. Fast convergence rates, in the order Op(NF−nrep/2), can be achieved, with NF the number of excited frequencies and nrep the number of repetitions of the GOBFs.

Takagi–Sugeno Fuzzy Models in the Framework of Orthonormal Basis Functions

An approach to obtain Takagi-Sugeno (TS) fuzzy models of nonlinear dynamic systems using the framework of orthonormal basis functions (OBFs) is presented in this paper. This approach is based on an architecture in which local linear models with ladder-structured generalized OBFs (GOBFs) constitute the fuzzy rule consequents and the outputs of the corresponding GOBF filters are input variables for the rule antecedents. The resulting GOBF-TS model is characterized by having only real-valued parameters that do not depend on any user specification about particular types of functions to be used in the orthonormal basis. The fuzzy rules of the model are initially obtained by means of a well-known technique based on fuzzy clustering and least squares. Those rules are then simplified, and the model parameters (GOBF poles, GOBF expansion coefficients, and fuzzy membership functions) are subsequently adjusted by using a nonlinear optimization algorithm. The exact gradients of an error functional with respect to the parameters to be optimized are computed analytically. Those gradients provide exact search directions for the optimization process, which relies solely on input-output data measured from the system to be modeled. An example is presented to illustrate the performance of this approach in the modeling of a complex nonlinear dynamic system.

QFT-based design of force and contact transition controllers for a flexible link manipulator

This paper deals with the modeling, system identification and robust control of flexible link manipulators that are required to perform contact task operations. For a single flexible link (SFL) manipulator in contact, two infinite dimensional models are developed and dynamic differences with respect to the force sensing devices are examined. Generalized orthonormal basis functions (GOBFs) are adopted for system identification and new algorithms are developed that improve the identification of resonant systems. The identification results, combined with estimated measures of model uncertainties, are directly used in the design of robust controllers. For the contact transition control, a switching condition is proposed based on robust position and force controllers. The stability of the switching controller is examined using a piecewise quadratic Lyapunov approach. Both simulation and experimental results are presented showing the effectiveness of the proposed technique.

Robust Predictive Control using a GOBF Model for MISO Systems

In this paper we develop a new method for robust predictive control for MISO systems represented on the Generalized Orthonormal Basis Functions. Unknown But Bounded Error approaches are used to update the uncertainty domain of the resultant model coefficients. This method uses a worst case strategy solved by a min-max optimization problem taking into account the constraints relative to parameter uncertainties and to measurement signals.

Application of orthonormal basis functions for identification of flexible-link manipulators

The advantage of generalized orthonormal bases, as compared to other orthonormal basis functions such as the Laguerre and Kautz functions, is the ability to include a priori knowledge of the real and complex pole locations of the model. In this paper, the implementation of generalized orthonormal basis functions (GOBF) for system identification is discussed and applied to identification of the dynamics of a single flexible-link manipulator. The plant is non-minimum phase with real and complex poles. A global optimization strategy for selecting the location of the poles of the basis functions is proposed. This method is evaluated through simulation and experimental studies and shows superior performance as compared to ARMAX and FIR identification.

Two-Stage and Projection Methods for Closed-Loop Identification Using Generalized Orthonormal Basis Functions

This paper proposes a closed-loop identification method based on the construction of generalized orthonormal basis functions (GOBF). It modifies the two-stage and projection methods, which apply a finite impulse (FIR) model structure to the estimation of the closed-loop sensitivity model in the input reconstruction step. In our approach the GOBF construction is applied instead of the FIR model in order to give better approximation of the sensitivity model. With appropriately selected basis functions in which the basis is generated by all-pass functions having poles close to the poles of the system the model structure is in a much more concise representation and the convergence rate can be faster. The paper gives the algorithm of the closed-loop identification method, and it demonstrates the method in a numerical example for an inverted pendulum device.

A N-Width Result for the Generalized Orthonormal Basis Function Model

In this paper we apply the Kolmogorov n-width concept to a model set based on the so-called generalized orthonormal basis functions (GOBFs), recently developed by Heuberger and co-workers. The main result of this paper is the characterization of the subset of H2(C+) for which the GOBFs are optimal in the Kolmogorov n-width sense. It turns out that each function of this subset is analytic outside a region of the complex plane that depends only on the poles of the GOBFs and on a parameter that defi1ws the size of the region of non-analyticity.

The Hambo Transform: A Signal and System Transform Induced by Generalized Orthonormal Basis Functions

A signal and system transformation is analyzed that is induced by a recently introduced generalized orthonormal basis for H2-systems and l2-signals. This basis is very flexible and generalizes the pulse, Laguerre and Kautz bases. The corresponding system and signal transformations generalize the Fourier and z-transforms; interesting properties of the representations in the transform domain are shown. The transformations are indispensable in the asymptotic analysis of related system identification algorithms, and provide powerful results in system approximation.

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

A least squares identification method is studied that estimates a finite number of expansion coefficients in the series expansion of a transfer function, where the expansion is in terms of recently introduced generalized basis functions. The basis functions are orthogonal in H2 and generalize the pulse, Laguerre and Kautz bases. One of their important properties is that when chosen properly they can substantially increase the speed of convergence of the series expansion. This leads to accurate approximate models with only few coefficients to be estimated. Explicit bounds are derived for the bias and variance errors that occur in the parameter estimates as well as in the resulting transfer function estimates.

Modelling Linear Dynamical Systems through Generalized Orthonormal Basis Functions

In many areas of signal, system and control theory orthogonal functions play an important role in issues of analysis and design. In this paper it is shown that there exist orthogonal functions that, in a natural way, are generated by stable linear dynamical systems, and that compose an orthonormal basis for the signal space l2n. To this end use is made of balanced realizations of inner transfer functions. The orthogonal functions can be considered as generalizations of e.g. the Laguerre functions and the pulse functions, related to the use of the delay operator, and give rise to an alternative series expansion of rational transfer functions. It is shown how we can exploit thcsp generalized basis functions to increase the speed of convergence in a series expansion, i.e. to obtain a good approximation by retaining only a finite number of expansion coefficients.