Identification of input-output LPV models

This chapter presents an overview of the available methods for identifying input-output LPV models both in discrete time and continuous time with the main focus on noise modeling issues. First, a least-squares approach and an instrumental variable method are presented for dealing with LPV-ARX models. Then, a refined instrumental variable approach is discussed to address more sophisticated noise models like Box-Jenkins in the LPV context. This latter approach is also introduced in continuous time and efficient solutions are proposed for both the problem of time-derivative approximation and the issue of continuous-time modeling of the noise

Refined instrumental variable estimation: maximum likelihood optimization of a unified Box–Jenkins model

For many years, various methods for the identification and estimation of parameters in linear, discretetime transfer functions have been available and implemented in widely available Toolboxes for MatlabTM. This paper considers a unified Refined Instrumental Variable (RIV) approach to the estimation of discrete and continuous-time transfer functions characterized by a unified operator that can be interpreted in terms of backward shift, derivative or delta operators. The estimation is based on the formulation of a pseudo-linear regression relationship involving optimal prefilters that is derived from an appropriately unified Box–Jenkins transfer function model. The paper shows that, contrary to apparently widely held beliefs, the iterative RIV algorithm provides a reliable solution to the maximum likelihood optimization equations for this class of Box–Jenkins transfer function models and so its en bloc or recursive parameter estimates are optimal in maximum likelihood, prediction error minimization and instrumental variable terms

Direct data-driven control of linear parameter-varying systems

In many control applications, nonlinear plants can be modeled as linear parameter-varying (LPV) systems, by which the dynamic behavior is assumed to be linear, but also dependent on some measurable signals, e.g., operating conditions. When a measured data set is available, LPV model identification can provide low complexity linear models that can embed the underlying nonlinear dynamic behavior of the plant. For such models, powerful control synthesis tools are available, but the way the modeling error and the conservativeness of the embedding affect the control performance is still largely unknown. Therefore, it appears to be attractive to directly synthesize the controller from data without modeling the plant. In this paper, a novel data-driven synthesis scheme is proposed to lay the basic foundations of future research on this challenging problem. The effectiveness of the proposed approach is illustrated by a numerical example

Nonlinear Model Predictive Controller Design for Identified Nonlinear Parameter Varying Model

In this paper, a novel nonlinear model predictive controller (MPC) is proposed based on an identified nonlinear parameter varying (NPV) model. First, an NPV model scheme is present for process identification, which is featured by its nonlinear hybrid Hammerstein model structure and varying model parameters. The hybrid Hammerstein model combines a normalized static artificial neural network with a linear transfer function to identify general nonlinear systems at each fixed working point. Meanwhile, a model interpolating philosophy is utilized to obtain the global model across the whole operation domain. The NPV model considers both the nonlinearity of transition dynamics due to the variation of the working-point and the nonlinear mapping from the input to the output at fixed working points. Moreover, under the new NPV framework, the control action is computed via a multistep linearization method aimed for nonlinear optimization problems. In the proposed scheme, only low cost tests are needed for system identification and the controller can achieve better output performance than MPC methods based on linear parameter varying (LPV) models. Numerical examples validate the effectiveness of the proposed approach

LPV model order selection in an LS-SVM setting

In parametric identification of Linear Parameter-Varying (LPV) systems, the scheduling dependencies of the model coefficients are commonly parameterized in terms of linear combinations of a-priori selected basis functions. Such functions need to be adequately chosen, e.g., on the basis of some first-principles or expert's knowledge of the system, in order to capture the unknown dependencies of the model coefficient functions on the scheduling variable and, at the same time, to achieve a low-variance of the model estimate by limiting the number of parameters to be identified. This problem together with the well-known model order selection problem (in terms of number of input lags, output lags and input delay of the model structure) in system identification can be interpreted as a trade-off between bias and variance of the resulting model estimate. The problem of basis function selection can be avoided by using a non-parametric estimator of the coefficient functions in terms of a recently proposed Least-Square Support-Vector-Machine (LS-SVM) approach. However, the selection of the model order still appears to be an open problem in the identification of LPV systems via the LS-SVM method. In this paper, we propose a novel reformulation of the LPV LS-SVM approach, which, besides of the non-parametric estimation of the coefficient functions, achieves data-driven model order selection via convex optimization. The properties of the introduced approach are illustrated via a simulation example