Identification of nonlinear interconnected systems

This thesis was submitted for the degree of Master of Philosophy and awarded by Brunel University.In this work we address the problem of identifying a discrete-time nonlinear system composed of a linear dynamical system connected to a static nonlinear component. We use linear fractional representation to provide a united framework for the identification of two classes of such systems. The first class consists of discrete-time systems consists of a linear time invariant system connected to a continuous nonlinear static component. The identification problem of estimating the unknown parameters of the linear system and simultaneously fitting a math order spline to the nonlinear data is addressed. A simple and tractable algorithm based on the separable least squares method is proposed for estimating the parameters of the linear and the nonlinear components. We also provide a sufficient condition on data for consistency of the identification algorithm. Numerical examples illustrate the performance of the algorithm. Further, we examine a second class of systems that may involve a nonlinear static element of a more complex structure. The nonlinearity may not be continuous and is approximated by piecewise a±ne maps defined on different convex polyhedra, which are defined by linear combinations of lagged inputs and outputs. An iterative identification procedure is proposed, which alternates the estimation of the linear and the nonlinear subsystems. Standard identification techniques are applied to the linear subsystem, whereas recently developed piecewise affine system identification techniques are employed for the estimation of the nonlinear component. Numerical examples show that the proposed procedure is able to successfully profit from the knowledge of the interconnection structure, in comparison with a direct black box identification of the piecewise a±ne system.Funding was obtained as a Marie Curie Early Stage Researcher Training fellowship, under the NET-ACE project (MEST-CT-2004-6724)

Identification of complex nonlinearities using cubic splines with automatic discretization

One of the major challenges in nonlinear system identification is the selection of appropriate mathematical functions to model the observed nonlinearities. In this context, piecewise polynomials, or splines, offer a simple and flexible representation basis requiring limited prior knowledge. The generally-adopted discretization for splines consists in an even distribution of their control points, termed knots. While this may prove successful for simple nonlinearities, a more advanced strategy is needed for nonlinear restoring forces with strong local variations. The present paper specifically introduces a two-step methodology to select automatically the location of the knots. It proposes to derive an initial model, using nonlinear subspace identification, and incorporating cubic spline basis functions with fixed and equally-spaced abscissas. In a second step, the location of the knots is optimized iteratively by minimizing a least-squares cost function. A single-degree-of-freedom system with a discontinuous stiffness characteristic is considered as a case study

Basis-function optimization for subspace-based nonlinear identification of systems with measured-input nonlinearities

Abstract-For nonlinear systems with measured-input nonlinearities, a subspace identification algorithm is used to identify the linear dynamics with the nonlinear mappings represented as a linear combination of basis functions. A selectiverefinement technique and a quasi-Newton optimization algorithm are used to iteratively improve the representation of the system nonlinearity. For both methods, polynomials, splines, sigmoids, wavelets, sines and cosines, or radial basis functions can be used as basis functions. Both approaches can be used to identify nonlinear maps with multiple arguments and with multiple outputs

A combined B-Spline-Neural-Network and ARX Model for Online Identi cation of Nonlinear Dynamic Actuation Systems

This paper presents a block oriented nonlinear dynamic model suitable for online identi cation.The model has the well known Hammerstein architecture where as a novelty the nonlinear static part is represented by a B-spline neural network (BSNN), and the linear static one is formalized by an auto regressive exogenous model (ARX). The model is suitable as a feed-forward control module in combination with a classical feedback controller to regulate velocity and position of pneumatic and hydraulic actuation systems which present non stationary nonlinear dynamics. The adaptation of both the linear and nonlinear parts is taking place simultaneously on a patterby- patter basis by applying a combination of error-driven learning rules and the recursive least squares method. This allows to decrease the amount of computation needed to identify the model's parameters and therefore makes the technique suitable for real time applications. The model was tested with a silver box benchmark and results show that the parameters are converging to a stable value after 1500 samples, equivalent to 7.5s of running time. The comparison with a pure ARX and BSNN model indicates a substantial improvement in terms of the RMS error, while the comparison with alternative non linear dynamic models like the NNOE and NNARX, having the same number of parameters but greater computational complexity, shows comparable performances

A combined B-Spline-Neural-Network and ARX Model for Online Identi cation of Nonlinear Dynamic Actuation Systems

This paper presents a block oriented nonlinear dynamic model suitable for online identi cation.The model has the well known Hammerstein architecture where as a novelty the nonlinear static part is represented by a B-spline neural network (BSNN), and the linear static one is formalized by an auto regressive exogenous model (ARX). The model is suitable as a feed-forward control module in combination with a classical feedback controller to regulate velocity and position of pneumatic and hydraulic actuation systems which present non stationary nonlinear dynamics. The adaptation of both the linear and nonlinear parts is taking place simultaneously on a patterby- patter basis by applying a combination of error-driven learning rules and the recursive least squares method. This allows to decrease the amount of computation needed to identify the model's parameters and therefore makes the technique suitable for real time applications. The model was tested with a silver box benchmark and results show that the parameters are converging to a stable value after 1500 samples, equivalent to 7.5s of running time. The comparison with a pure ARX and BSNN model indicates a substantial improvement in terms of the RMS error, while the comparison with alternative non linear dynamic models like the NNOE and NNARX, having the same number of parameters but greater computational complexity, shows comparable performances

Modeling of complex-valued Wiener systems using B-spline neural network

In this brief, a new complex-valued B-spline neural network is introduced in order to model the complex-valued Wiener system using observational input/output data. The complex-valued nonlinear static function in the Wiener system is represented using the tensor product from two univariate Bspline neural networks, using the real and imaginary parts of the system input. Following the use of a simple least squares parameter initialization scheme, the Gauss–Newton algorithm is applied for the parameter estimation, which incorporates the De Boor algorithm, including both the B-spline curve and the first-order derivatives recursion. Numerical examples, including a nonlinear high-power amplifier model in communication systems, are used to demonstrate the efficacy of the proposed approaches

Diabetes Mellitus Modelling Based on Blood Glucose Measurements

Insulin Dependant Diabetes Mellitus(IDDM) is a chronic disease characterized by the inability of the pancreas to produce sufficient amounts of insulin. To cover the deficiency 4-6 insulin injections have to be taken daily. The aim of this insulin therapy is to maintain normoglycemia, blood glucose level between 4-7 mmol/L. To determine the amount and timing of these injections different approaches are used. Mostly qualitative and semiquantitative models and reasoning are used to design such a therapy. In this Master Thesis an attempt is made to show how system identification and automatic control perspectives may be used to estimate quantitative models. Such models can then be used to design optimal insulin regimens. The system was divided into three subsystems, the insulin subsystem, the glucose subsystem and the insulin/glucose interaction. The insulin subsystem aims to describe the absorption of injected insulin from the subcutaneous depots and the glucose subsystem the absorption of glucose from the gut following a meal. These subsystems were modelled using compartment models and proposed models found in the literature. Several black box models and grey-box models describing the insulin/glucose interaction have been developed and analysed. These models have been fitted to real data monitored by a IDDM patient. Many difficulties were encountered, typical of biomedical systems. Non-uniform and scarce sampling, time-varying dynamics and severe non-linearities were some of the difficulties encountered during the modelling. None of the proposed models were able to describe the system accurately. However, all the linear models shared some dynamics, and there is ground to suspect that these dynamics are essential parts of the true system. More research has to be undertaken, primarily to investigate the non-linear nature of the system and to see whether other variables than glucose flux and insulin absorption are important for the dynamics of the system

Modeling and identification of nonlinear systems using SISO LEM-Hammerstein and LEM-Wiener model structures

This paper applies the concept of linearization around the equilibrium manifold (LEM) already presented in the literature in order to construct model structures that can be viewed as extensions of the conventional Wiener and Hammerstein models. Instead of linear time-invariant subsystems in association with static nonlinearities, these extensions exhibit variable dynamic character and can therefore model a broader class of systems than the conventional cited approaches. Moreover, the identification strategy already used with LEM systems can be applied in order to construct such models from experiments, and the techniques destined for analysis and control of Wiener and Hammerstein systems can be applied promptly. To application of these concepts to the modeling and identification is demonstrated with a numerical example, considering a heat exchange system