P. Lopes dos Santos

Identification of Bilinear Systems Using an Iterative Deterministic-Stochastic Subspace Approach

In this paper we introduce a new identification algorithm for MIMO bilinear systems driven by white noise inputs. The new algorithm is based on a convergent sequence of linear deterministic-stochastic state space approximations, thus considered a Picard based method. The key to the algorithm is the fact that the bilinear terms behave like white noise processes. Using a linear Kalman filter, the bilinear terms can be estimated and combined with the system inputs at each iteration, leading to a linear system which can be identified with a linear-deterministic subspace algorithm such as MOESP, N4SID, or CVA. Furthermore, the model parameters obtained with the new algorithm converge to those of a bilinear model. Finally, the dimensions of the data matrices are comparable to those of a linear subspace algorithm, thus avoiding the curse of dimensionality.

Identification of LPV systems using successive approximations

In this paper a successive approximation approach for MIMO linear parameter varying (LPV) systems with affine parameter dependence is proposed. This new approach is based on an algorithm previously introduced by the authors, which elaborates on a convergent sequence of linear deterministic-stochastic state-space approximations. In the previous algorithm the bilinear term between the time varying parameter vector and the state vector is allowed to behave as a white noise process when the scheduling parameter is a white noise sequence. However, this is a strong limitation in practice since, most often than not, the scheduling parameter is imposed by the process itself and it is typically a non white noise signal. In this paper, the bilinear term is analysed for non white noise scheduling sequences. It is concluded that its behaviour depends on the input sequence itself and it ranges from acting as an independent colored noise source, mostly removed by the identification algorithm, down to a highly input correlated signal that may be incorrectly assumed as being part of the system subspace. Based on the premise that the algorithm performance can be improved by the noise energy reduction, the bilinear term is expressed as a function of past inputs, scheduling parameters, outputs, and states, and the linear terms are included in a new extended input.

LPV system identification using a separable least squares support vector machines approach

In this article, an algorithm to identify LPV State Space models for both continuous-time and discrete-time systems is proposed. The LPV state space system is in the Companion Reachable Canonical Form. The output vector coefficients are linear combinations of a set of a possibly infinite number of nonlinear basis functions dependent on the scheduling signal, the state matrix is either time invariant or a linear combination of a finite number of basis functions of the scheduling signal and the input vector is time invariant. This model structure, although simple, can describe accurately the behaviour of many nonlinear SISO systems by an adequate choice of the scheduling signal. It also partially solves the problems of structural bias caused by inaccurate selection of the basis functions and high variance of the estimates due to over-parameterisation. The use of an infinite number of basis functions in the output vector increases the flexibility to describe complex functions and makes it possible to learn the underlying dependencies of these coefficients from the data. A Least Squares Support Vector Machine (LS-SVM) approach is used to address the infinite dimension of the output coefficients. Since there is a linear dependence of the output on the output vector coefficients and, on the other hand, the LS-SVM solution is a nonlinear function of the state and input matrix coefficients, the LPV system is identified by minimising a quadratic function of the output function in a reduced parameter space; the minimisation of the error is performed by a separable approach where the parameters of the fixed matrices are calculated using a gradient method. The derivatives required by this algorithm are the output of either an LTI or an LPV (in the case of a time-varying SS matrix) system, that need to be simulated at every iteration. The effectiveness of the algorithm is assessed on several simulated examples.

Parameter estimation of discrete and continuous-time physical models: A similarity transformation approach

The fitting of physical dynamical models to stimulus-response data such as the chemical concentration measured after a gas has been released to the environment, or the plasma concentration measured after an intravenous or oral input of a drug, are important problems in the area of system identification. Using models of different structures, one can obtain relevant statistical information on the parameters of the model from an array of software packages available in the literature. A meaningful interpretation of these parameters requires that in the presence of error-free data and an error-free model structure, a unique solution for the model parameters is guaranteed. This problem is known as a priori identifiability. Once the model is deemed identifiable, the parameters are then obtained, usually via a nonlinear least squares technique. In addition to identifiability, there is the problem of convergence of the parameters to the true values. It is a known fact that nonlinear parameter estimation algorithms do not always converge to the true parameter set. This is due to the fact that estimating the parameters of a nonlinear model can at times be an ill-conditioned problem. In this paper we use the same state space analysis techniques used to determine identifiability, to estimate the model parameters in a linear fashion. We approach the problem from a system identification point of view and then take advantage of the similarity transformation between the physical model and the identified model. We formulate the similarity relations and then transform them into a null space problem whose solution leads to the physical parameters. The novelty of our approach is in the use of a state space system identification algorithm to identify a black-box system, followed by a physical parameter extraction step using robust numerical tools such as the singular value decomposition.

Identification of a Benchmark Wiener-Hammerstein System by Bilinear and Hammerstein-Bilinear Models

Abstract In this paper the Wiener-Hammerstein system proposed as a benchmark for the SYSID 2009 benchmark session is identified as a bilinear discrete system. The bilinear approximation relies on both facts that the Wiener-Hammerstein system can be described by a Volterra series which can be approximated by bilinear systems. The identification is performed with an iterative bilinear subspace identification algorithm previously proposed by the authors. In order to increase accuracy, polynomial static nonlinearities are added to the bilinear model input. These Hammerstein type bilinear models are then identified using the same iterative subspace identification algorithm.

PDE Model for Leakage Detection in High Pressure Gas Networks

In this paper we design a model based method to locate a leakage and estimate its size in a gas network, using a linearised version of an hyperbolic PDE. To do this, the problem is reduced to two identical ODEs, allowing in this way for a representation of the pressure as well as the mass flow in terms of its system of fundamental solutions. Then using the available measurements at the grid boundary points, the correspondent coefficients can be determined. Assuming pressure continuity, we check for consistency of the coefficients in order to find faulty pipelines. Thence, the location of the leakage can be found either graphically or using a numerical method for a specific pipe. Next, its size can also be estimated.

Indirect continuous-time system identification—A subspace downsampling approach

This article presents a new indirect identification method for continuous-time systems able to resolve the problem of fast sampling. To do this, a Subspace IDentification Down-Sampling (SIDDS) approach that takes into consideration the intermediate sampling instants of the input signal is proposed. This is done by partitioning the data set into m subsets, where m is the downsampling factor. Then, the discrete-time model is identified using a based subspace identification discrete-time algorithm where the data subsets are fused into a single one. Using the algebraic properties of the system, some of the parameters of the continuous-time model are directly estimated. A procedure that secures a prescribed number of zeros for the continuous-time model is used during the estimation process. The algorithms performance is illustrated through an example of fast sampling, where its performance is compared with the direct methods implemented in Contsid.

A lumped transfer function model for High Pressure Gas Pipelines

In this paper a lumped transfer function (TF) model is derived for High Pressure Natural Gas Pipelines. Departing from a nonlinear partial differential equation (PDE) model a high order continuous state space (SS) linear model is obtained using a finite difference method. An infinite order TF is calculated from the SS representation and finally is approximated by a compact non-rational function. This model is compared with SIMONE®, a commercial simulator of gas transport and distribution, using a case study, and both exhibit a similar accuracy.

Gas pipelines LPV modelling and identification for leakage detection

A new approach to gas leakage detection in high pressure distribution networks is proposed, where the pipeline is modelled as a Linear Parameter Varying (LPV) System driven by the source node mass flow with the pressure as the scheduling parameter, and the system output as the mass flow at the offtake. Using a recently proposed successive approximations LPV system subspace identification algorithm, the pipeline is thus identified from operational data. The leak is detected using a Kalman filter where the fault is treated as an augmented state. The effectiveness of this method is illustrated with an example with a mixture of real and simulated data.

A new insight to the matrices extraction in a MOESP type subspace identification algorithm

In this paper we analyse the estimates of the matrices produced by the non-biased deterministic-stochastic subspace identification algorithms (NBDSSI) proposed by Van Overschee and De Moor (1996). First, an alternate expression is derived for the A and C estimates. It is shown that the Chiuso and Picci result (Chiuso and Picci 2004) stating that the A and C estimates delivered by this algorithm robust version and by the Verhaegens MOESP (Verhaegen and Dewilde 1992a, Verhaegen and Dewilde 1992b, Verhaegen 1993, Verhaegen 1994) are equal, can be obtained from this expression. An alternative approach for the estimation of matrices B and D in subspace identification is also described. It is shown that the least squares approach for the estimation of these matrices estimation can be just expressed as an orthogonal projection of the future outputs on a lower dimension subspace in the orthogonal complement of the column space of the extended observability matrix. Since this subspace has a dimension equal to the number of outputs, a simpler and numerically more efficient (but equally accurate) new subspace algorithm is provided.