Pb Pepijn Cox

LPV system identification under noise corrupted scheduling and output signal observations

Most of the approaches available in the literature for the identification of Linear Parameter-Varying (LPV) systems rely on the assumption that only the measurements of the output signal are corrupted by the noise, while the observations of the scheduling variable are considered to be noise free. However, in practice, this turns out to be an unrealistic assumption in most of the cases, as the scheduling variable is often related to a measured signal and, thus, it is inherently affected by a measurement noise. In this paper, it is shown that neglecting the noise on the scheduling signal, which corresponds to an error-in-variables problem, can lead to a significant bias on the estimated parameters. Consequently, in order to overcome this corruptive phenomenon affecting practical use of data-driven LPV modeling, we present an identification scheme to compute a consistent estimate of LPV Input/Output (IO) models from noisy output and scheduling signal observations. A simulation example is provided to prove the effectiveness of the proposed methodology.

Bayesian identification of LPV Box-Jenkins models

In this paper, we introduce a nonparametric approach in a Bayesian setting to efficiently estimate, both in the stochastic and computational sense, linear parameter-varying (LPV) input-output models under general noise conditions of Box-Jenkins (BJ) type. The approach is based on the estimation of the one-step-ahead predictor model of general LPV-BJ structures, where the sub-predictors associated with the input and output signals are captured as asymptotically stable infinite impulse response models (IIRs). These IIR sub-predictors are identified in a completely nonparametric sense, where not only the coefficients are estimated as functions, but also the whole time evolution of the impulse response is estimated as a function. In this Bayesian setting, the one-step-ahead predictor is modelled as a zero-mean Gaussian random field, where the covariance function is a multidimensional Gaussian kernel that encodes both the possible structural dependencies and the stability of the predictor. The unknown hyperparameters that parameterize the kernel are tuned using the empirical Bayes approach, i.e., optimization of the marginal likelihood with respect to available data. It is also shown that, in case the predictor has a finite order, i.e., the true system has an ARX noise structure, our approach is able to recover the underlying structural dependencies. The performance of the identification method is demonstrated on LPV-ARX and LPV-BJ simulation examples by means of a Monte Carlo study.

CVA identification of nonlinear systems with LPV state-space models of affine dependence

This paper discusses an improvement on the extension of linear subspace methods (originally developed in the Linear Time-Invariant (LTI) context) to the identification of Linear Parameter-Varying (LPV) and state-affine nonlinear system models. This includes the fitting of a special polynomial shifted form based LPV Autoregressive with eXogenous input (ARX) model to the observed input-output data. The estimated ARX model is used for filtering away the effects of future inputs on future outputs to obtain the so called “corrected future” analogous to the LTI case. The generality of the applied LPV-ARX parametrization now permits the estimation of the input-output map of a rather general class of LPV state-space models with matrices depending affinely on the scheduling. This is achieved by a canonical variate analysis (CVA) between the past and the corrected future which provides an estimate of a relevant set of state variables and their trajectories for the system, necessary for the construction of the minimal order state equations.

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.

LPV State-space model identification in the Bayesian setting: A 3-step procedure

Current state-of-the-art linear parameter-varying (LPV) control design methods presume that an LPV state-space (SS) model of the system with affine dependence on the scheduling variable is available. However, many existing LPV-SS identification schemes either suffer heavily from computational issues related to the curse of dimensionality or are based on severe approximations. To overcome these issues, in this paper, the Bayesian framework is combined with a recently developed efficient SS realization scheme. We propose a computationally attractive 3-step approach for identifying LPV-SS models. In Step 1, the sub-Markov parameters representing the impulse response of the system are estimated in a Bayesian setting, using kernel based Ridge regression with hyper-parameter tuning via marginal likelihood optimization. Subsequently, in Step 2, an LPV-SS realization is obtained by using an efficient basis reduced Ho-Kalman like deterministic SS realization scheme on the identified impulse response. Finally, in Step 3, to reach the maximum likelihood estimate, the LPV-SS model is refined by applying a Bayesian expectation-maximization method. The performance of the proposed 3-step scheme is demonstrated on a Monte-Carlo simulation study.