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A Duality Approach to System Identification of LTV and NL Systems

Joe A. Roman-FloresSupervisor: Dr. Paolo Rapisarda

University of Southampton

UKACC PhD Presentation Showcase

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Introduction

Ø Identification theory for linear systems has strong basis and well-establish methodologies.

Ø Identification for nonlinear systems without prior information it is hard to achieve.

Ø Using duality approach allow us to obtain a state model from data, and the data will give us internal information about the system.

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Approach

Ø Based on input-output data (𝑢, 𝑦)we construct state trajectories 𝑥corresponding to it.

Ø Identify a state model from such input-state-output trajectories.

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Duality

Primal Dual

Ø The duality approach for identification exploits the duality relation between the external trajectories and the internal ones.

External : input-output 𝒖, 𝒚 &(𝒖+,𝒚′)

Internal : state 𝒙 &(𝒙+)

𝑑𝑑𝒕 𝒙 0 = 𝑨 0 𝒙 0 + 𝑩 0 𝒖 0𝒚 0 = 𝑪 0 𝒙 0

𝑑𝑑𝒕 𝒙

+ 0 = −𝑨 0 7𝒙+ 0 + 𝑪 0 7𝒖+ 0

𝒚+ 0 = 𝑩 0 7𝒙′ 0

LTV systems

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Identification procedure

Ø Procedure• Produce N experiments to get (𝑢, 𝑦)and (𝑢′, 𝑦′).• Using the duality relation obtain a matrix of data𝐸 𝑡 ∈ ℝ<×< representing the internal properties.

• Factorize the state from this matrix.• Obtain a set of equations to obtain a state model.

𝑢+7𝑦 − 𝑦+7𝑢 =𝑑𝑑𝑡 (𝑥

+7𝑄𝑥)

external internal

𝐸(𝑡)

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Identification procedureSelf-dual systems

Ø Matrix of data E(t) has special properties that allows to obtain information of the internal in a natural way.

• Easy factorization of the state.• State dimension from data.

Nonlinear systemsØ Identification using the variational system.

𝑢7𝑦− 𝑦7𝑢 =𝑑𝑑𝑡 (𝑥

7𝑄𝑥)

external internal

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Issues to overcome

Ø From numbers to functions.Ø Sufficiently informative data.Ø Factorization.

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