Abstract

In this paper, we establish a unified framework for subspace identification (SID) of linear parameter-varying (LPV) systems to estimate LPV state–space (SS) models in innovation form. This framework enables us to derive novel LPV SID schemes that are extensions of existing linear time-invariant (LTI) methods. More specifically, we derive the open-loop, closed-loop, and predictor-based data-equations (input–output surrogate forms of the SS representation) by systematically establishing an LPV subspace identification theory. We also show the additional challenges of the LPV setting compared to the LTI case. Based on the data-equations, several methods are proposed to estimate LPV-SS models based on a maximum-likelihood or a realization based argument. Furthermore, the established theoretical framework for the LPV subspace identification problem allows us to lower the number of to-be-estimated parameters and to overcome dimensionality problems of the involved matrices, leading to a decrease in the computational complexity of LPV SIDs in general. To the authors’ knowledge, this paper is the first in-depth examination of the LPV subspace identification problem. The effectiveness of the proposed subspace identification methods are demonstrated and compared with existing methods in a Monte Carlo study of identifying a benchmark MIMO LPV system.

Highlights

  • Realization based state-space identification techniques, so-called subspace identification (SID) methods, have been successfully applied in practice to estimate timevarying and/or nonlinear dynamical systems using linear parameter-varying (LPV) state-space (SS) models

  • We establish a unified framework for subspace identification (SID) of linear parameter-varying (LPV) systems to estimate LPV state-space (SS) models in innovation form

  • The existing techniques are based on predictor based subspace identification (PBSID) [8], past-output multivariable output-error state-space (PO-MOESP) [3], canonical variate analysis (CVA) [7], or the successive approximation identification algorithm [4]

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Summary

Introduction

Realization based state-space identification techniques, so-called subspace identification (SID) methods, have been successfully applied in practice to estimate timevarying and/or nonlinear dynamical systems using linear parameter-varying (LPV) state-space (SS) models. The goal of this paper is to obtain a unified formulation to treat the LPV subspace identification problem and derive its associated stochastic properties by systematically establishing an LPV SID theory This unified framework enables us to (i) understand relations and performance of LPV SIDs, (ii) extend most of the successful LTI subspace schemes to the LPV setting, (iii) decrease the dimensionality problems, and (iv) relax assumptions on the scheduling signal. The state realization problem is tackled based both on a maximum-likelihood and a realization argument This is acomplished first for the open-loop (Sec. 4) and for the closed-loop identification setting (Sec. 5), leading to the LPV formulation of various well-known LTI subspace methods. The efficiency of the unified framework is demonstrated by a Monte Carlo study on an LPV-SS identification benchmark (Sec. 6)

The LPV data-equations
The data-generating system
The open-loop data-equation
The generalized expectation operation Eof a process u is defined as
Derivation of the predictor
Parametric subspace identification setting
Subspace identification in open-loop form
Main concept
Maximum-likelihood estimation
Realization based estimation
Subspace identification in closed-loop form
Simulation Example
Identification setting
Analysis of the results
Conclusion
A Proof of Theorem 1
Full Text
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