Identification of parametric models in the frequency-domain through the subspace framework under LMI constraints

International audienceIn this paper, an algorithm to identify parametric systems with an affine (or polynomial) parameter dependence through the subspace framework is proposed. It stands as an extension of the standard subspace-based algorithm which is well established in the Linear Time Invariant (LTI) case. The formulation is close to the LTI identification scheme and simply involves frequency-domain data obtained at different operating points (the parameters are frozen during each experiment). The proposed algorithm allows to identify directly a parameter-dependent model instead of interpolating multiple local models as in traditional local approaches. Another contribution is that it is possible to impose the poles location through Linear Matrix Inequalities (LMI) constraints, extending what has been done in the LTI case. This technique is applied to a numerical example and to real industrial frequency-domain data originating from an open-channel flow simulation for hydroelectricity production

Predictive LPV control of a liquid-gas separation process

[EN] The problem of controlling a liquid-gas separation process is approached by using LPV control techniques. An LPV model is derived from a nonlinear model of the process using differential inclusion techniques. Once an LPV model is available, an LPV controller can be synthesized. The authors present a predictive LPV controller based on the GPC controller [Clarke D, Mohtadi C, Tuffs P. Generalized predictive control - Part I. Automatica 1987;23(2):137-48; Clarke D, Mohtadi C, Tuffs P. Generalized predictive control - Part II. Extensions and interpretations. Automatica 1987;23(2):149-60]. The resulting controller is denoted as GPC-LPV. This one shows the same structure as a general LPV controller [El Gahoui L, Scorletti G. Control of rational systems using linear-fractional representations and linear matrix inequalities. Automatica 1996;32(9):1273-84; Scorletti G, El Ghaoui L. Improved LMI conditions for gain scheduling and related control problems. International Journal of Robust Nonlinear Control 1998;8:845-77; Apkarian P, Tuan HD. Parametrized LMIs in control theory. In: Proceedings of the 37th IEEE conference on decision and control; 1998. p. 152-7; Scherer CW. LPV control and full block multipliers. Automatica 2001;37:361-75], which presents a linear fractional dependence on the process signal measurements. Therefore, this controller has the ability of modifying its dynamics depending on measurements leading to a possibly nonlinear controller. That controller is designed in two steps. First, for a given steady state point is obtained a linear GPC using a linear local model of the nonlinear system around that operating point. And second, using bilinear and linear matrix inequalities (BMIs/LMIs) the remaining matrices of GPC-LPV are selected in order to achieve some closed loop properties: stability in some operation zone, norm bounding of some input/output channels, maximum settling time, maximum overshoot, etc., given some LPV model for the nonlinear system. As an application, a GPC-LPV is designed for the derived LPV model of the liquid-gas separation process. This methodology can be applied to any nonlinear system which can be embedded in an LPV system using differential inclusion techniques. (C) 2006 Elsevier Ltd. All rights reserved.Partially supported by projects: CICYT DPI2004-08383-C03-02 and DPI2005-07835.Salcedo-Romero-De-Ávila, J.; Martínez Iranzo, MA.; Ramos Fernández, C.; Herrero Durá, JM. (2007). Predictive LPV control of a liquid-gas separation process. Advances in Engineering Software. 38(7):466-474. https://doi.org/10.1016/j.advengsoft.2006.10.003S46647438

Towards Efficient Maximum Likelihood Estimation of LPV-SS Models

How to efficiently identify multiple-input multiple-output (MIMO) linear parameter-varying (LPV) discrete-time state-space (SS) models with affine dependence on the scheduling variable still remains an open question, as identification methods proposed in the literature suffer heavily from the curse of dimensionality and/or depend on over-restrictive approximations of the measured signal behaviors. However, obtaining an SS model of the targeted system is crucial for many LPV control synthesis methods, as these synthesis tools are almost exclusively formulated for the aforementioned representation of the system dynamics. Therefore, in this paper, we tackle the problem by combining state-of-the-art LPV input-output (IO) identification methods with an LPV-IO to LPV-SS realization scheme and a maximum likelihood refinement step. The resulting modular LPV-SS identification approach achieves statical efficiency with a relatively low computational load. The method contains the following three steps: 1) estimation of the Markov coefficient sequence of the underlying system using correlation analysis or Bayesian impulse response estimation, then 2) LPV-SS realization of the estimated coefficients by using a basis reduced Ho-Kalman method, and 3) refinement of the LPV-SS model estimate from a maximum-likelihood point of view by a gradient-based or an expectation-maximization optimization methodology. The effectiveness of the full identification scheme is demonstrated by a Monte Carlo study where our proposed method is compared to existing schemes for identifying a MIMO LPV system

On the Consistency of Certain Identification Methods for Linear Parameter Varying Systems

The consistency of certain identification methods for Linear Parameter Varying systems is considered. More precisely, methods for the identification of SISO input-output models are analysed. In order to perform a consistency analysis the application of ergodicity is required, which is not obviously applicable with these types of time-varying systems. It is therefore shown that, when the scheduling parameter satisfies certain conditions, ergodicity type results can be applied to the methods considered. An analysis is then carried out for two cases: that of noise-free measurements of the scheduling parameter, and then the more realistic case of noisy scheduling parameter measurements. The latter is shown to be an errors-in-variables type problem. In both cases the least squares technique is shown to typically give biased estimates and the instrumental variables method is proposed as a way of resolving this. The analysis carried out is reinforced by results in simulation

Data-driven Precompensator Tuning for Linear Parameter Varying Systems

Methods for direct data-driven tuning of the parameters of precompensators for LPV systems are developed. Since the commutativity property is not always satisfied for LPV systems, previously proposed methods for LTI systems that use this property cannot be directly adapted. When the inverse of the system exists in the proposed parameterisation of the precompensator, the LPV transfer functions commute and an algorithm using only two experiments on the real system is proposed. It is shown that this algorithm gives consistent estimates of the parameters of the system inverse despite the presence of stochastic disturbances. For the more general case, when the system inverse does not belong to the set of parameterised precompensators, another technique is developed. This technique requires a number of experiments equal to twice the number of precompensator parameters and it is shown that the calculated parameters minimise the mean squared tracking error

Realization Theory for LPV State-Space Representations with Affine Dependence

In this paper we present a Kalman-style realization theory for linear parameter-varying state-space representations whose matrices depend on the scheduling variables in an affine way (abbreviated as LPV-SSA representations). We deal both with the discrete-time and the continuous-time cases. We show that such a LPV-SSA representation is a minimal (in the sense of having the least number of state-variables) representation of its input-output function, if and only if it is observable and span-reachable. We show that any two minimal LPV-SSA representations of the same input-output function are related by a linear isomorphism, and the isomorphism does not depend on the scheduling variable.We show that an input-output function can be represented by a LPV-SSA representation if and only if the Hankel-matrix of the input-output function has a finite rank. In fact, the rank of the Hankel-matrix gives the dimension of a minimal LPV-SSA representation. Moreover, we can formulate a counterpart of partial realization theory for LPV-SSA representation and prove correctness of the Kalman-Ho algorithm. These results thus represent the basis of systems theory for LPV-SSA representation.Comment: The main difference with respect to the previous version is as follows: typos have been fixe

A convex relaxation approach to set-membership identication of LPV systems

Identification of linear parameter varying models is considered in the paper, under the assumption that both the output and the scheduling parameter measurements are affected by bounded noise. First, the problem of computing parameter uncertainty intervals is formulated in terms of nonconvex optimization. Then, on the basis of the analysis of the regressor structure, we present an ad hoc convex relaxation scheme to compute parameter bounds by means of semidefinite optimization

Identification of Multimodel LPV Models with Asymmetric Gaussian Weighting Function

This paper is concerned with the identification of linear parameter varying (LPV) systems by utilizing a multimodel structure. To improve the approximation capability of the LPV model, asymmetric Gaussian weighting functions are introduced and compared with commonly used symmetric Gaussian functions. By this mean, locations of operating points can be selected freely. It has been demonstrated through simulations with a high purity distillation column that the identified models provide more satisfactory approximation. Moreover, an experiment is performed on real HVAC (heating, ventilation, and air-conditioning) to further validate the effectiveness of the proposed approach

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