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The latter means for example that the model structure is either well applicable in a gain-scheduling type of approach, i. The basis functions, that provide bases for the space H2 Hilbert space of complex functions that are squared integrable on the unit circle , are generated by a cascaded network of stable all-pass filters, whose pole locations represent the prior knowledge about the system at hand.
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This implies that every transfer function Fp z that corresponds to 1. In LTI identification, only a finite number of terms in 1. In contrast with FIR structures, the OBF parametrization can achieve almost zero modeling error with a relatively small 18 1 Introduction number of parameters, due to the infinite impulse-response characteristics of the basis functions. Then, based on experiments with constant p, frozen aspects of 1.
Due to the linearity of 1. In the prediction-error framework, they can be considered with an output-error type of noise model. The resulting model structure has a linear-in-the-parameter property, which implies that its estimation is available through linear regression. Non-asymptotic variance and bias bounds of the estimates are also available.
These fruitful properties imply that direct identification of LPV systems may be beneficial in terms of the model structure 1. However, for the investigation of this approach, first an LPV prediction-error framework needs to be established. This means that a wide class of nonlinear systems can be identified with such OBF model 1. It is obvious that 1. Based on the general approximator property and the attractive identification properties of OBF models, successful identification approaches based on these structures have been introduced in the NL and the fuzzy field see [63, ].
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These methods provide low complexity and reliable estimates for the considered classes of systems, giving the hope that similar mechanisms could also be successfully applied in the LPV case. It is well known in the LTI case that the approximation error, i. This refers back to the observation that model structure selection has a prime importance in the LPV case, and as for other type of LPV models, this choice influences the achievable maximal accuracy of the estimates.
In conclusion, the above discussed perspectives yield the observation that identification of LPV systems with OBF models like 1. Based on this observation, we aim to develop an identification framework in this book where relations of model structures and the concept of estimation can be understood.
In fact we intend to extend the prediction-error framework to the LPV case and investigate how we cab benefit from the classical results. Additionally, we also intend to develop an effective LPV identification mechanism, based on the promising properties of OBF expansion models, that overcomes the drawbacks of the existing state-of-the-art solutions see Sect.
However, it has been revealed in the previous section that the main obstacle in the formulation of such an identification framework originates from the gaps of LPV system theory. To fill these gaps and give a unified system theoretical framework in the LPV case, which enables the investigation and comparison of model structures and also provides an algebraic tool for manipulations on them, a new system theoretical framework, which is based on the extension of the LTI behavioral approach , is worked out in the first part of the book.
In the sequel we present a theory with new ideas of LPV behaviors, modeling, and identification methods, that have appealing properties and can be applied successfully to fulfill the aimed objectives. Due to the vast number of problems this book addresses, we also leave many open ends and questions for future research. The exploration of all issues of the LPV identification cycle of general physical systems remains a problem, but the use of orthonormal basis functions, the behavioral 20 1 Introduction theory, and the proposed prediction-error identification framework open interesting and promising perspectives to approach the identification of this system class efficiently.
In Chap. In this framework it becomes possible to understand relations of LPV models and the behavioral approach also gives the basic tool to introduce OBFs-based LPV model parameterizations later on. The discovered relations provide the tools to analyze and compare LPV model structures. It is shown that finite truncations of these representations can be used as model structures for LPV system identification. Such structures also have wide approximation capabilities.
Discretization of LPV systems is reviewed in Chap. New discretization approaches are introduced together with criteria to choose the discretization step size. Using the framework of the LPV behavioral approach, a new mechanism is introduced that solves the LPV modeling issue of such systems. This approach together with the discretization methods of Chap. The method is based on the clustering of sample poles that result from identification of the system with constant scheduling signals.
The effect of noise on identification is also considered and a robust basis-selection procedure is developed based on hyperbolic-geometry results. The prediction-error framework for the LPV case is established and the model structures of the current approaches are analyzed. Two identification approaches, a global and a local one, are formulated with static dependence of the coefficients.
The former approach utilizes the gain-scheduling-based interpolation concept while the latter yields a global estimate based on linear regression. To overcome the limiting assumption of static dependence needed for the parametrization of these approaches, two alternative OBFs-based model structures are worked out.
These structures enable the approximation of dynamic dependence of the coefficients through a feedback with only static dependence. An identification approach of such feedback structures is derived through a separable least-squares strategy. This chapter is devoted to the introduction of basic concepts that are fundamental for the theory developed in the subsequent chapters.
The notion of orthonormal basis functions OBFs is introduced with a brief overview of the most important relations and properties. Next, the basic concepts and theory of LTI system identification are reviewed focusing on prediction-error methods. OBFs related parameterizations and the associated identification approaches are also introduced together with the optimality concept of OBF model structures in terms of the Kolmogorov n-width theory.
The most simplest of these expansions is the Laurent expansion. Assume that F is stable, so the domain of F z is the exterior of the unit circle. Thus 2. Note that such a representation is available for unstable systems as well, but in terms of the forward-shift operator q. Finite truncations of the IRR are known as finite impulse response FIR models, which have proven their usefulness in many areas of engineering, ranging from signal processing to control design and also in approximative system identification. Despite numerous attractive properties of FIR models, these structures often require a large number of expansion terms to adequately approximate the original system dynamics.
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Such models preserve all the advantages of FIR structures, but in general they require much less expansion terms for adequate approximation due to the infinite impulse response IIR characteristics of the basis functions. To simplify the following discussion, which is based on [73, ] and , we restrict our attention to discrete time DT stable SISO systems, however some hints are also given later how the theory extends to MIMO or continuous time CT systems.
Also introduce E to represent the exterior of J. The function space we frequently use in the sequel is the following: Definition 2. H2 E is equipped with the following inner product: Definition 2. We distinguish a special set of functions in H2 E , the so-called inner functions. However, note that 2. Basically, the class of OBFs generalized by 2. These categories, which hierarchically contain the smaller classes from pulse basis to the Takenaka-Malmquist class, are defined as follows see  for a detailed overview on these classes : 2.
For a particular3 balanced realization of the inner functions, it can be shown that the basis functions generated via 2. We allow here SS representations, i.
police-risk-management.com/order/putting/dov-android-spia-cellulare.php Note that a balanced state-space realization of a inner function is non-unique. However, due to 2. In the following we use the M1AB based construction if not indicated otherwise. Such a basis is commonly considered to be optimal for F. Such basis sequences can be considered to be adequate for the expansion of systems with dominating second order modes. Based on the expansion of the transfer functions of LTI dynamical systems in terms of the pulse basis, the impulse response representation of LTI systems is available.
The generated functions provide a series-expansion 2. However, the degree of freedom in the basis selection is much higher in the MIMO case. Note, that it is possible to use different basis sequences in the generation of 2. The derivation is the same as the state-space construction approach presented by 2.
An important issue here is the construction of the square allpass function Gb z. Where in the scalar case, Gb z can be written as a Blaschke product and thus modulo the sign it is determined by the poles of the product, the multivariable-case is more involved and inhibits more freedom. There is however a completely analogous theory for CT systems, where the Laplace transform is used instead of the Z-transform.
The two most important steps of this cycle are the choices of an appropriate model set and the identification criterion. While the previous describes the set in which the suitable description of the system is sought, the latter defines the aimed performance of the model. The choice of the model set is crucial as it directly influences the maximum achievable accuracy or quality of the identified model in terms of the user-defined criterion. The model set should be as large as possible in order to contain as many candidate models as possible, which reduces the structural or bias error of the optimal model in the set.