From Nonlinear Identification to Linear Parameter Varying Models: Benchmark Examples

Linear parameter-varying (LPV) models form a powerful model class to analyze and control a (nonlinear) system of interest. Identifying a LPV model of a nonlinear system can be challenging due to the difficulty of selecting the scheduling variable(s) a priori, which is quite challenging in case a first principles based understanding of the system is unavailable. This paper presents a systematic LPV embedding approach starting from nonlinear fractional representation models. A nonlinear system is identified first using a nonlinear block-oriented linear fractional representation (LFR) model. This nonlinear LFR model class is embedded into the LPV model class by factorization of the static nonlinear block present in the model. As a result of the factorization a LPV-LFR or a LPV state-space model with an affine dependency results. This approach facilitates the selection of the scheduling variable from a data-driven perspective. Furthermore the estimation is not affected by measurement noise on the scheduling variables, which is often left untreated by LPV model identification methods. The proposed approach is illustrated on two well-established nonlinear modeling benchmark examples

On the Simulation of Polynomial NARMAX Models

In this paper, we show that the common approach for simulation non-linear stochastic models, commonly used in system identification, via setting the noise contributions to zero results in a biased response. We also demonstrate that to achieve unbiased simulation of finite order NARMAX models, in general, we require infinite order simulation models. The main contributions of the paper are two-fold. Firstly, an alternate representation of polynomial NARMAX models, based on Hermite polynomials, is proposed. The proposed representation provides a convenient way to translate a polynomial NARMAX model to a corresponding simulation model by simply setting certain terms to zero. This translation is exact when the simulation model can be written as an NFIR model. Secondly, a parameterized approximation method is proposed to curtail infinite order simulation models to a finite order. The proposed approximation can be viewed as a trade-off between the conventional approach of setting noise contributions to zero and the approach of incorporating the bias introduced by higher-order moments of the noise distribution. Simulation studies are provided to illustrate the utility of the proposed representation and approximation method.Comment: Accepted in IEEE CDC 201

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

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

A Comparison of LPV Gain Scheduling and Control Contraction Metrics for Nonlinear Control

Gain-scheduled control based on linear parameter-varying (LPV) models derived from local linearizations is a widespread nonlinear technique for tracking time-varying setpoints. Recently, a nonlinear control scheme based on Control Contraction Metrics (CCMs) has been developed to track arbitrary admissible trajectories. This paper presents a comparison study of these two approaches. We show that the CCM based approach is an extended gain-scheduled control scheme which achieves global reference-independent stability and performance through an exact control realization which integrates a series of local LPV controllers on a particular path between the current and reference states.Comment: IFAC LPVS 201

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

Data-driven LPV modeling of continuous pulp digesters

In this technical report, the LPV-IO identification techniques described in Kauven et al. [2013] (Chapter 5) are applied in order to estimate an LPV model of a continuous pulp digester. The pulp digester simulator (described in Modén [2011]) has been provided by ABB for benchmark studies as part of its participation in the EU project Autoprofi

An SDP approach for l0-minimization: application to ARX model segmentation

Abstract Minimizing the ℓ 0 -seminorm of a vector under convex constraints is a combinatorial (NP-hard) problem. Replacement of the ℓ 0 -seminorm with the ℓ 1 -norm is a commonly used approach to compute an approximate solution of the original ℓ 0 -minimization problem by means of convex programming. In the theory of compressive sensing, the condition that the sensing matrix satisfies the Restricted Isometry Property (RIP) is a sufficient condition to guarantee that the solution of the ℓ 1 -approximated problem is equal to the solution of the original ℓ 0 -minimization problem. However, the evaluation of the conservativeness of the ℓ 1 -relaxation approaches is recognized to be a difficult task in case the {RIP} is not satisfied. In this paper, we present an alternative approach to minimize the ℓ 0 -norm of a vector under given constraints. In particular, we show that an ℓ 0 -minimization problem can be relaxed into a sequence of semidefinite programming problems, whose solutions are guaranteed to converge to the optimizer (if unique) of the original combinatorial problem also in case the {RIP} is not satisfied. Segmentation of {ARX} models is then discussed in order to show, through a relevant problem in system identification, that the proposed approach outperforms the ℓ 1 -based relaxation in detecting piece-wise constant parameter changes in the estimated model

Segmentation of ARX systems through SDP-relaxation techniques

Segmentation of ARX models can be formulated as a combinato- rial minimization problem in terms of the ℓ0-norm of the param- eter variations and the ℓ2-loss of the prediction error. A typical approach to compute an approximate solution to such a prob- lem is based on ℓ1-relaxation. Unfortunately, evaluation of the level of accuracy of the ℓ1-relaxation in approximating the opti- mal solution of the original combinatorial problem is not easy to accomplish. In this poster, an alternative approach is proposed which provides an attractive solution for the ℓ0-norm minimiza- tion problem associated with segmentation of ARX models