Joe A. Roman-Flores Supervisor: Dr. Paolo Rapisarda...
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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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