Abstract
In this article, we introduce an iterative subspace system identification algorithm for MIMO linear parameter-varying systems with innovation-type noise models driven by general inputs and a measurable white noise time-varying parameter vector. The new algorithm is based on a convergent sequence of linear deterministic-stochastic state-space approximations, thus considered a Picard-based method. Such methods have proven to be convergent for the bilinear state-space system identification problem. Their greatest strength lies on the dimensions of the data matrices that are comparable to those of a linear subspace algorithm, thus avoiding the curse of dimensionality.

Abstract
The class of subspace system identification algorithms is used here to derive new identification algorithms for 2-D causal, recursive, and separable-in-denominator (CRSD) state space systems in the Roesser form. The algorithms take a known deterministic input-output pair of 2-D signals and compute the system order (n) and system parameter matrices {A, B, C, D}. Since the CRSD model can be treated as two 1-D systems, the proposed algorithms first separate the vertical component from the state and output equations and then formulate a set of 1-D horizontal subspace equations. The solution to the horizontal subproblem contains all the information necessary to compute (n) and {A, B, C, D}. Four algorithms are presented for the identification of CRSD models directly from input-output data: an intersection algorithm, (N4SID), (MOESP), and (CCA). The intersection algorithm is distinguished from the rest in that it computes the state sequences, as well as the system parameters, whereas N4SID, MOESP, and CCA differ primarily in the way they compute the system parameter matrices {A1, C1}. The advantage of the intersection algorithm is that the identified model is in balanced coordinates, thus ideally suited for 2-D model reduction. However, it is computationally more expensive than the other algorithms. A comparison of all algorithms is presented.

Abstract
In this paper a new approach to gas leakage detection in high pressure natural gas transportation networks is proposed. The pipeline is modelled as a Linear Parameter Varying (LPV) System driven by the source node massflow with the gas inventory variation in the pipe (linepack variation, proportional to the pressure variation) as the scheduling parameter. The massflow at the offtake node is taken as the system output. The system is identified by the Successive Approximations LPV System Subspace Identification Algorithm which is also described in this paper. The leakage is detected using a Kalman filter where the fault is treated as an augmented state. Given that the gas linepack can be estimated from the massflow balance equation, a differential method is proposed to improve the leakage detector effectiveness. A small section of a gas pipeline crossing Portugal in the direction South to North is used as a case study. LPV models are identified from normal operational data and their accuracy is analyzed. The proposed LPV Kalman filter based methods are compared with a standard mass balance method in a simulated 10% leakage detection scenario. The Differential Kalman Filter method proved to be highly efficient.

Abstract
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.

Abstract
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(R), a commercial simulator of gas transport and distribution, using a case study, and both exhibit a similar accuracy.

Abstract
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.