Psc Peter Heuberger

Asymptotically optimal orthonormal basis functions for LPV system identification

A global model structure is developed for parametrization and identification of a general class of Linear Parameter-Varying (LPV) systems. By using a fixed orthonormal basis function (OBF) structure, a linearly parametrized model structure follows for which the coefficients are dependent on a scheduling signal. An optimal set of OBFs for this model structure is selected on the basis of local linear dynamic properties of the LPV system (system poles) that occur for different constant scheduling signals. The selected OBF set guarantees in an asymptotic sense the least worst-case modeling error for any local model of the LPV system. Through the fusion of the Kolmogorov n-width theory and Fuzzy c-Means clustering, an approach is developed to solve the OBF-selection problem for discrete-time LPV systems, based on the clustering of observed sample system poles.

The Behavioral Approach to Linear Parameter-Varying Systems

Linear parameter-varying (LPV) systems are usually described in either state-space or input-output form. When analyzing system equivalence between different representations it appears that the time-shifted versions of the scheduling signal (dynamic dependence) need to be taken into account. Therefore, representations used previously to define and specify LPV systems are not equal in terms of dynamics. In order to construct a parametrization-free description of LPV systems that overcomes these difficulties, a behavioral approach is introduced that serves as a basis for specifying system theoretic properties. LPV systems are defined as the collection of trajectories of system variables (like inputs and outputs) and scheduling variables. LPV kernel, input-output, and state-space system representations are introduced with appropriate equivalence transformations.

Order and structural dependence selection of LPV-ARX models using a nonnegative garrote approach

In order to accurately identify Linear Parameter-Varying (LPV) systems, order selection of LPV linear regression models has prime importance. Existing identification approaches in this context suffer from the drawback that a set of functional dependencies needs to be chosen a priori for the parametrization of the model coefficients. However in a black-box setting, it has not been possible so far to decide which functions from a given set are required for the parametrization and which are not. To provide a practical solution, a nonnegative garrote approach is applied. It is shown that using only a measured data record of the plant, both the order selection and the selection of structural coefficient dependence can be solved by the proposed method.

Discretization of Linear Fractional Representations of LPV systems

Commonly, controllers for linear parameter-varying (LPV) systems are designed in continuous time using a linear fractional representation (LFR) of the plant. However, the resulting controllers are implemented on digital hardware. Furthermore, discrete-time LPV synthesis approaches require a discrete-time model of the plant which is often derived from a continuous-time first-principle model. Existing discretization approaches for LFRs describing LPV systems suffer from disadvantages like the possibility of serious approximation errors, issues of complexity, etc. To explore the disadvantages, existing discretization methods are reviewed and novel approaches are derived to overcome them. The proposed and existing methods are compared and analyzed in terms of approximation error, considering ideal zero-order hold actuation and sampling. Criteria to choose appropriate sampling times with respect to the investigated methods are also presented. The proposed discretization methods are tested and compared both on a simulation example and on the electronic throttle control problem of a race motorcycle.

LPV system identification with globally fixed orthonormal basis functions

A global and a local identification approach are developed for approximation of linear parameter-varying (LPV) systems. The utilized model structure is a linear combination of globally fixed (scheduling-independent) orthonormal basis functions (OBFs) with scheduling-parameter dependent weights. Whether the weighting is applied on the input or on the output side of the OBFs, the resulting models have different modeling capabilities. The local identification approach of these structures is based on the interpolation of locally identified LTI models on the scheduling domain where the local models are composed from a fixed set of OBFs. The global approach utilizes a priori chosen functional dependence of the parameter-varying weighting of a fixed set of OBFs to deliver global model estimation from measured I/O data. Selection of the OBFs that guarantee the least worst-case modeling error for the local behaviors in an asymptotic sense, is accomplished through the fuzzy Kolmogorov c-max approach. The proposed methods are analyzed in terms of applicability and consistency of the estimates.

Predictor input selection for direct identification in dynamic networks

In the literature methods have been proposed which enable consistent estimates of modules embedded in complex dynamic networks. In this paper the network extension of the so called closed-loop Direct Method is investigated. Currently, for this method the variables which must be included in the predictor model are not considered as a user choice. In this paper it is shown that there is some freedom as to which variables to include in the predictor model as inputs, and still obtain consistent estimates of the module of interest. Conditions on this choice of predictor inputs are presented.

An LPV identification Framework Based on Orthonormal Basis Functions

Describing nonlinear dynamic systems by Linear Parameter-Varying (LPV) models has become an attractive tool for control of complicated systems with regime-dependent (linear) behavior. For the identification of LPV models from experimental data a number of methods has been presented in the literature but a full picture of the underlying identification problem is still missing. In this contribution a solid system theoretic basis for the description of model structures for LPV systems is presented, together with a general approach to the LPV identification problem. Use is made of a series-expansion approach, employing orthogonal basis functions.

Flexible model structures for LPV identification with static scheduling dependency

A discrete-time linear parameter-varying (LPV) model can be seen as the combination of local LTI models together with a scheduling signal dependent function set, that selects one of the models to describe the continuation of the signal trajectories at every time instant. An identification strategy of LPV models is proposed that consists of the separate approximation of the local model set and the scheduling functions. The local model set is represented as a linear combination (series expansion) of orthonormal basis functions (OBFs). The expansion coefficients are dynamically dependent (weighting) functions of the scheduling parameters (depending on time shifted scheduling). To approximate this dependency class with a static one (non-shifted scheduling), a feedback-based structure of the weighting functions is introduced. The proposed model structure is identified in a two step procedure. First the OBFs, that guarantee the least asymptotic worst-case modeling error for the local models, are selected through the fuzzy Kolmogorov c-Max approach. With the resulting OBFs, the weighting functions are identified through a separable least-squares algorithm. The method is demonstrated by means of simulation examples and analyzed in terms of applicability, convergence, and consistency of the model estimates.

Orthonormal basis selection for LPV system identification, the Fuzzy-Kolmogorov c-Max approach

A fuzzy clustering approach is developed to select pole locations for orthonormal basis functions (OBFs), used for identification of linear parameter varying (LPV) systems. The identification approach is based on interpolation of locally identified linear time invariant (LTI) models, using globally fixed OBFs. Selection of the optimal OBF structure, that guarantees the least worst-case local modelling error in an asymptotic sense, is accomplished through the fusion of the Kolmogorov n-width (KnW) theory and fuzzy c-means (FcM) clustering of observed sample system poles

Optimal pole selection for LPV system identification with OBFs, a clustering approach

A fuzzy clustering approach is studied for optimal pole selection of Orthonormal Basis Functions (OBFs) used for the identification of Linear Parameter Varying (LPV) systems. The identification approach is based on interpolation of locally identified Linear Time Invariant (LTI) models, using globally fixed OBFs. The selection of the optimal OBF structure, that guarantees the least worst-case local modelling error, is accomplished through the joint application of the Kolmogorov n-width theory and Fuzzy c-Means (FCM) clustering of observed sample system poles. For the problem at hand, FCM solutions are given, based on three different metrics, and the qualities of the results are compared in terms of the derived OBFs.