Giuseppe Giordano and Jonas Sj oberg Department of Signals and - PowerPoint PPT Presentation

Giuseppe Giordano and Jonas Sj¨ oberg Department of Signals and Systems Maximum Likelihood identification of Wiener-Hammerstein model in presence of process noise Chalmers University of Technology

Outline 2 / 16 Motivation 1 Identification in presence of process noise 2 Illustrative examples 3 Maximum Likelihood identification 4 Benchmark results 5

Motivation 3 / 16 Introduction The Wiener-Hammerstein model in presence of process noise Only signals u and y available for identification ( N data points). Noises w , e zero mean and normally distributed.

Motivation 3 / 16 Introduction The Wiener-Hammerstein model in presence of process noise Only signals u and y available for identification ( N data points). Noises w , e zero mean and normally distributed. When w ( t ) ≡ 0 (only measurement noise):

Motivation 3 / 16 Introduction The Wiener-Hammerstein model in presence of process noise Only signals u and y available for identification ( N data points). Noises w , e zero mean and normally distributed. When w ( t ) ≡ 0 (only measurement noise): The Best Linear Approximation (BLA) is a consistent estimate of concatenation of the linear parts: G BLA = kG 1 G 2

Motivation 3 / 16 Introduction The Wiener-Hammerstein model in presence of process noise Only signals u and y available for identification ( N data points). Noises w , e zero mean and normally distributed. When w ( t ) ≡ 0 (only measurement noise): The Best Linear Approximation (BLA) is a consistent estimate of concatenation of the linear parts: G BLA = kG 1 G 2 Main problem: splitting of the dynamics and non-linearity estimation

Motivation 3 / 16 Introduction The Wiener-Hammerstein model in presence of process noise Only signals u and y available for identification ( N data points). Noises w , e zero mean and normally distributed. When w ( t ) ≡ 0 (only measurement noise): The Best Linear Approximation (BLA) is a consistent estimate of concatenation of the linear parts: G BLA = kG 1 G 2 Main problem: splitting of the dynamics and non-linearity estimation Several solutions to the splitting problem in literature based on BLA

Motivation 3 / 16 Introduction The Wiener-Hammerstein model in presence of process noise Only signals u and y available for identification ( N data points). Noises w , e zero mean and normally distributed. When w ( t ) ≡ 0 (only measurement noise): The Best Linear Approximation (BLA) is a consistent estimate of concatenation of the linear parts: G BLA = kG 1 G 2 Main problem: splitting of the dynamics and non-linearity estimation Several solutions to the splitting problem in literature based on BLA Question: Can we use BLA in presence of process noise? How can we implement the split?

Identification in presence of process noise 4 / 16 The Best Linear Approximation in case of process noise Data generated from a Wiener-Hammerstein system y ( t ) = G 0 2 ( q ) f ( x ( t )) + e ( t ) (1a) x ( t ) = x 0 ( t ) + w ( t ) (1b) x 0 ( t ) = G 0 1 ( q ) u ( t ) (1c)

Identification in presence of process noise 4 / 16 The Best Linear Approximation in case of process noise Data generated from a Wiener-Hammerstein system y ( t ) = G 0 2 ( q ) f ( x ( t )) + e ( t ) (1a) x ( t ) = x 0 ( t ) + w ( t ) (1b) x 0 ( t ) = G 0 1 ( q ) u ( t ) (1c) Theorem (Consistency of the BLA) Signals u ( t ) , w ( t ) , e ( t ) independent, Gaussian, stationary processes. G ( q , θ ) arbitrary transfer function parametrization, with freely adjustable gain. Parameter θ is estimated from u and y using an output error method, N 1 ˆ � ( y ( t ) − G ( q , θ ) u ( t )) 2 , θ N = argmin (2) N θ t =1 then G ( q , ˆ θ N ) → kG 0 1 ( q ) G 0 2 ( q ) w.p. 1 as N → ∞ (3)

Identification in presence of process noise 5 / 16 The splitting problem The BLA is still a consistent estimate of the linear parts. Can we use a standard splitting algorithm? How will the estimation of the non-linearity look like?

Identification in presence of process noise 5 / 16 The splitting problem The BLA is still a consistent estimate of the linear parts. Can we use a standard splitting algorithm? How will the estimation of the non-linearity look like? A standard splitting algorithm based on the BLA ( exhaustive search ), when w ( t ) ≡ 0:

Identification in presence of process noise 5 / 16 The splitting problem The BLA is still a consistent estimate of the linear parts. Can we use a standard splitting algorithm? How will the estimation of the non-linearity look like? A standard splitting algorithm based on the BLA ( exhaustive search ), when w ( t ) ≡ 0: Consider all possible combinations of the poles/zeros of the BLA.

Identification in presence of process noise 5 / 16 The splitting problem The BLA is still a consistent estimate of the linear parts. Can we use a standard splitting algorithm? How will the estimation of the non-linearity look like? A standard splitting algorithm based on the BLA ( exhaustive search ), when w ( t ) ≡ 0: Consider all possible combinations of the poles/zeros of the BLA. For each combination estimate the non-linearity, using PEM.

Identification in presence of process noise 5 / 16 The splitting problem The BLA is still a consistent estimate of the linear parts. Can we use a standard splitting algorithm? How will the estimation of the non-linearity look like? A standard splitting algorithm based on the BLA ( exhaustive search ), when w ( t ) ≡ 0: Consider all possible combinations of the poles/zeros of the BLA. For each combination estimate the non-linearity, using PEM. When w ( t ) ≡ 0, PEM is equivalent to the Maximum likelihood estimator (ML), hence the estimate is consistent.

Identification in presence of process noise 5 / 16 The splitting problem The BLA is still a consistent estimate of the linear parts. Can we use a standard splitting algorithm? How will the estimation of the non-linearity look like? A standard splitting algorithm based on the BLA ( exhaustive search ), when w ( t ) ≡ 0: Consider all possible combinations of the poles/zeros of the BLA. For each combination estimate the non-linearity, using PEM. When w ( t ) ≡ 0, PEM is equivalent to the Maximum likelihood estimator (ML), hence the estimate is consistent. If α i , β i is a possible split of the dynamics in the BLA, PEM estimate is N 1 � ( y ( t ) − G 2 ( q , β i ) f ( G 1 ( q , α i ) u ( t ) , γ )) 2 . γ ∗ = argmin (4) N γ t =1

Identification in presence of process noise 5 / 16 The splitting problem The BLA is still a consistent estimate of the linear parts. Can we use a standard splitting algorithm? How will the estimation of the non-linearity look like? A standard splitting algorithm based on the BLA ( exhaustive search ), when w ( t ) ≡ 0: Consider all possible combinations of the poles/zeros of the BLA. For each combination estimate the non-linearity, using PEM. When w ( t ) ≡ 0, PEM is equivalent to the Maximum likelihood estimator (ML), hence the estimate is consistent. If α i , β i is a possible split of the dynamics in the BLA, PEM estimate is N 1 � ( y ( t ) − G 2 ( q , β i ) f ( G 1 ( q , α i ) u ( t ) , γ )) 2 . γ ∗ = argmin (4) N γ t =1 One estimation will provide the true split (lowest prediction error).

Identification in presence of process noise 5 / 16 The splitting problem The BLA is still a consistent estimate of the linear parts. Can we use a standard splitting algorithm? How will the estimation of the non-linearity look like? A standard splitting algorithm based on the BLA ( exhaustive search ), when w ( t ) ≡ 0: Consider all possible combinations of the poles/zeros of the BLA. For each combination estimate the non-linearity, using PEM. When w ( t ) ≡ 0, PEM is equivalent to the Maximum likelihood estimator (ML), hence the estimate is consistent. If α i , β i is a possible split of the dynamics in the BLA, PEM estimate is N 1 � ( y ( t ) − G 2 ( q , β i ) f ( G 1 ( q , α i ) u ( t ) , γ )) 2 . γ ∗ = argmin (4) N γ t =1 One estimation will provide the true split (lowest prediction error). Consistency of (4) needed both for the estimation of non-linearity and for the splitting algorithm.

Identification in presence of process noise 6 / 16 Impact of the process noise on PEM Problem: The PEM estimate (4) is not consistent in presence of process noise

Identification in presence of process noise 6 / 16 Impact of the process noise on PEM Problem: The PEM estimate (4) is not consistent in presence of process noise Outcomes:

Identification in presence of process noise 6 / 16 Impact of the process noise on PEM Problem: The PEM estimate (4) is not consistent in presence of process noise Outcomes: Even assuming the true split of the dynamics is known, the estimation of the non-linearity will be biased.

Identification in presence of process noise 6 / 16 Impact of the process noise on PEM Problem: The PEM estimate (4) is not consistent in presence of process noise Outcomes: Even assuming the true split of the dynamics is known, the estimation of the non-linearity will be biased. In the exhaustive search algorithm, a wrong split could provide lowest prediction error. The splitting fails.

Illustrative examples 7 / 16 First order systems with polynomial non-linearity Data points (1000) generated from the true system: q x 0 ( t ) = q − a 1 u ( t ) z ( t ) = f ( x 0 ( t ) + w ( t )) (5) q y ( t ) = q − a 2 z ( t ) + e ( t ) f ( x ) is a third degree polynomial: f ( x ( t )) = c 0 + c 1 x ( t ) + c 2 x 2 ( t ) + c 3 x 3 ( t ) (6)