### Abstract

We present a new class of models, called uncertain-input models, that allows us to treat system-identification problems in which a linear system is subject to a partially unknown input signal. To encode prior information about the input or the linear system, we use Gaussian-process models. We estimate the model from data using the empirical Bayes approach: the hyperparameters that characterize the Gaussian-process models are estimated from the marginal likelihood of the data. We propose an iterative algorithm to find the hyperparameters that relies on the EM method and results in decoupled update steps. Because in the uncertain-input setting neither the marginal likelihood nor the posterior distribution of the unknowns is tractable, we develop an approximation approach based on variational Bayes. As part of the contribution of the paper, we show that this model structure encompasses many classical problems in system identification such as Hammerstein models, blind system identification, and cascaded linear systems. This connection allows us to build a systematic procedure that applies effectively to all the aforementioned problems, as shown in the numerical simulations presented in the paper.

Language | English |
---|---|

Pages | 130-141 |

Number of pages | 12 |

Journal | Automatica |

Volume | 105 |

DOIs | |

State | Published - 1 Jul 2019 |

### Fingerprint

### Keywords

- Estimation algorithms
- Gaussian processes
- Nonlinear models
- Nonparametric identification
- System identification

### Cite this

*Automatica*,

*105*, 130-141. DOI: 10.1016/j.automatica.2019.03.014

}

*Automatica*, vol. 105, pp. 130-141. DOI: 10.1016/j.automatica.2019.03.014

**Modeling and identification of uncertain-input systems.** / Risuleo, Riccardo Sven (Corresponding author); Bottegal, Giulio; Hjalmarsson, Håkan.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Modeling and identification of uncertain-input systems

AU - Risuleo,Riccardo Sven

AU - Bottegal,Giulio

AU - Hjalmarsson,Håkan

PY - 2019/7/1

Y1 - 2019/7/1

N2 - We present a new class of models, called uncertain-input models, that allows us to treat system-identification problems in which a linear system is subject to a partially unknown input signal. To encode prior information about the input or the linear system, we use Gaussian-process models. We estimate the model from data using the empirical Bayes approach: the hyperparameters that characterize the Gaussian-process models are estimated from the marginal likelihood of the data. We propose an iterative algorithm to find the hyperparameters that relies on the EM method and results in decoupled update steps. Because in the uncertain-input setting neither the marginal likelihood nor the posterior distribution of the unknowns is tractable, we develop an approximation approach based on variational Bayes. As part of the contribution of the paper, we show that this model structure encompasses many classical problems in system identification such as Hammerstein models, blind system identification, and cascaded linear systems. This connection allows us to build a systematic procedure that applies effectively to all the aforementioned problems, as shown in the numerical simulations presented in the paper.

AB - We present a new class of models, called uncertain-input models, that allows us to treat system-identification problems in which a linear system is subject to a partially unknown input signal. To encode prior information about the input or the linear system, we use Gaussian-process models. We estimate the model from data using the empirical Bayes approach: the hyperparameters that characterize the Gaussian-process models are estimated from the marginal likelihood of the data. We propose an iterative algorithm to find the hyperparameters that relies on the EM method and results in decoupled update steps. Because in the uncertain-input setting neither the marginal likelihood nor the posterior distribution of the unknowns is tractable, we develop an approximation approach based on variational Bayes. As part of the contribution of the paper, we show that this model structure encompasses many classical problems in system identification such as Hammerstein models, blind system identification, and cascaded linear systems. This connection allows us to build a systematic procedure that applies effectively to all the aforementioned problems, as shown in the numerical simulations presented in the paper.

KW - Estimation algorithms

KW - Gaussian processes

KW - Nonlinear models

KW - Nonparametric identification

KW - System identification

UR - http://www.scopus.com/inward/record.url?scp=85063903295&partnerID=8YFLogxK

U2 - 10.1016/j.automatica.2019.03.014

DO - 10.1016/j.automatica.2019.03.014

M3 - Article

VL - 105

SP - 130

EP - 141

JO - Automatica

T2 - Automatica

JF - Automatica

SN - 0005-1098

ER -