Jan Goos

Estimation of Linear Parameter-Varying affine state space models using synchronized periodic input and scheduling signals

During the past decades some very interesting results have been obtained in controller synthesis using Linear Parameter-Varying (LPV) systems. However, the LPV models are commonly required to be transformed into State Space (SS) form. We tackle the LPV SS identification problem directly in the frequency domain. To the best of our knowledge, this is a novel approach. When the input and scheduling are chosen to be periodic and synchronized, the state space equations are structured and sparse in the frequency domain. The parameters of these state space equations are estimated by minimizing a weighted non-linear least squares criterion. Starting values are generated via the Best Linear Time-Invariant (BLTI) approximation. The resulting model is also valid for non-periodic scheduling and input signals.

Realization and identification of autonomous linear periodically time-varying systems☆

Abstract The subsampling of a linear periodically time-varying system results in a collection of linear time-invariant systems with common poles. This key fact, known as “lifting”, is used in a two-step realization method. The first step is the realization of the time-invariant dynamics (the lifted system). Computationally, this step is a rank-revealing factorization of a block-Hankel matrix. The second step derives a state space representation of the periodic time-varying system. It is shown that no extra computations are required in the second step. The computational complexity of the overall method is therefore equal to the complexity for the realization of the lifted system. A modification of the realization method is proposed, which makes the complexity independent of the parameter variation period. Replacing the rank-revealing factorization in the realization algorithm by structured low-rank approximation yields a maximum likelihood identification method. Existing methods for structured low-rank approximation are used to identify efficiently a linear periodically time-varying system. These methods can deal with missing data.

Frequency domain weighted nonlinear least squares estimation of parameter-varying differential equations

This paper presents a frequency domain identification technique for estimation of Linear Parameter-Varying (LPV) differential equations. In a band-limited setting, it is shown that the time derivatives of the input and output signals can be computed exactly in the frequency domain, even for non-periodic inputs and parameter variations. The method operates in an errors-in-variables framework (noisy input and output), but the scheduling signal is assumed to be known. Under these conditions, the proposed estimator is proven to be consistent.

Continuous-time identification of periodically parameter-varying state space models

This paper presents a new frequency domain identification technique to estimate multivariate Linear Parameter-Varying (LPV) continuous-time state space models, where a periodic variation of the parameters is assumed or imposed. The main goal is to obtain an LPV state space model suitable for control, from a single parameter-varying experiment. Although most LPV controller synthesis tools require continuous time state space models, the identification of such models is new. The proposed identification method designs a periodic input signal, taking the periodicity of the parameter variation into account. We show that when an integer number of periods is observed for both the input and the scheduling, the state space model representation has a specific, sparse structure in the frequency domain, which is exploited to speed up the estimation procedure. A weighted non-linear least squares algorithm then minimizes the output error. Two initialization methods are explored to generate starting values. The first approach uses a Linear Time-Invariant (LTI) approximation. The second estimates a Linear Time-Variant (LTV) input-output differential equation, from which a corresponding state space realization is computed.

Minimal state space realisation of continuous-time linear time-variant input–output models

ABSTRACT In the linear time-invariant (LTI) framework, the transformation from an input–output equation into state space representation is well understood. Several canonical forms exist that realise the same dynamic behaviour. If the coefficients become time-varying however, the LTI transformation no longer holds. We prove by induction that there exists a closed-form expression for the observability canonical state space model, using binomial coefficients.

Comparative study of two global affine Linear Periodic Parameter Varying State Space model estimation algorithms

Abstract A comparative study is made between two global Linear Periodic Parameter-Varying (LPPV) identification algorithms. The first method is a state-of-the-art subspace identification method in the time domain. The second is a newly developed frequency domain approach, where the identification experiment is designed carefully so we can exploit the resulting structure. For both methods, the result is a state space model with an affine dependence on the varying parameters, which can be used for LPV control synthesis. Simulations show that the frequency domain procedure has a lower variance for identical experimental conditions.

Generalizing periodically time-varying measurements with a parameter-varying input-output model

In many real-life systems, the dynamic behavior varies periodically in time, as a function of an external scheduling parameter. Although it is possible to capture these changing dynamics with a periodically time-varying model, no prediction can be made for a new trajectory of the time variation. In this paper, we extract a more general (non-parametric) parameter-varying model, from a set of periodically time-varying measurements on a parameter-varying bandpass filter.

Estimation of affine LPV state space models in the frequency domain: Extension to transient behavior and non-periodic inputs

Recently, a state space identification method was developed for linear periodic parameter-varying systems in the frequency domain. The proposed algorithm required the input and scheduling signals to be periodic and synchronized. In this paper, we extend the existing approach to handle transient behavior with respect to the input. Because the system is no longer required to be in steady state, non-periodic inputs can now be used as well. We illustrate the extended identification technique on a discrete time SIMO simulation example.

Detection and quantification of dynamic dependence in linear parameter-varying differential equations

In the identification of linear parameter-varying models, it is usually assumed that the model coefficients vary instantaneously with the external scheduling parameter. Any dynamics in the model coefficients with respect to the scheduling are therefore neglected. In this paper, we propose a method to detect and quantify the dynamic dependence of the coefficient functions. Proceeding in this way, it can be verified whether the scheduling dynamics are indeed negligible. First, a general linear time-varying model is identified. Subsequently, the covariance of the estimated model parameters is calculated, which allows the computation of a confidence bound for the coefficient functions. Eventually, it is determined whether the dynamic dependence on the scheduling parameter falls in this confidence bound and, thus, whether it is justifiable to use a model with a static dependence on the scheduling parameter.

Continuous time frequency domain LPV state space identification via periodic time-varying input-output modeling

We aim to identify a parameter-varying state space model that is suited for control design. Current LPV controller synthesis tools usually require a state space formulation that is affine in the scheduling parameters. We therefore present a frequency domain state space identification method for periodic parameter variation, in continuous time. First, we identify a periodic time-varying input-output differential equation. Next, this representation is transformed into a time-varying state space form. We use a closed-form expression for the states, consisting of binomial coefficients and derivatives of the original differential equation coefficients. Finally, an affine LPV state space model is fitted. The difficulty is to select the proper basis functions, but in this routine, we have an educated guess. Special attention is given to the sparsity and structure in the frequency domain calculations.