Adaptive input design for ARX systems

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Adaptive input design for ARX systems
László Gerencsér MTA SZTAKI Hungarian Academy of
Science Budapest, Hugary Håkan
Hjalmarsson School of Electrical
Engineering KTH Stockholm, Sweden
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Input Design for Identification
Leads to optimization problems that are
typically non-convex and infinite-dimensional,
and dependent on true system
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Outline

• A typical input design problem
• Formulation
• Mathematical formulation
• Convexification
• Finite dimensionalization
• Characteristics
• History
• Adaptive input design
• Basic idea
• Parameter convergence
• Asymptotic distribution
• Binary signals
• Numerical illustration
• Summary

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Prediction Error Identification

• Given input/output data from the linear process
• estimate a model, using a
  Prediction Error criterion

• Assumptions
• Variance errors only G0G(q)
• open loop operation

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A Typical Input Design Problem

• Quantity of interest J(G0)

Examples Static gain G0(1),
Impulse response coefficient,
Zero, Pole.
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Mathematical Formulation
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Convexification
Schur complement)
Convex in ?ubut infinite dimensional problem
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Finite dimensionalization
But now infinite dimensional constraint in
?! KYP) LMI constraint
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Input design. Summary

• Performance constraint LMI
• Spectrum positivity constraint LMI
• Free variables hc0,...,cm,auxiliary variables
  X

General optimal input design problem (SDP)
minc,X V(c,X,?) s.t. M(c,X,?) 0
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Open loop input design

• Cooley and Lee Control-relevant experiment
  design for multivariable systems described by
  expansions in orthonormal bases, Automatica 01.
• Hildebrand and Gevers (2003) Identification for
  control Optimal input design with respect to a
  worst case \nu-gap function,SICON
• Jansson (2004) Experiment Design with
  Applications in Identification for Control, PhD
  thesis, KTH.
• Jansson and Hjalmarsson (2005) Input Design via
  LMIs Admitting Frequency-wise Model
  Specifications in Confidence Regions, IEEE Trans.
  Automatic Control 2005,
• Bombois, Scorletti, Gevers, Hildebrand and Van
  den Hof Least costly identification experiment
  for control, Automatica 07.
• Evolution of work by Goodwin and Payne, Zarrop,
  Mehra, Söderström
• LMI formulations
• Wide class of objective functions and model
  quality constraints
• Frequency-by-frequency constraints
• Quality specifications in confidence ellipsoids

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Adaptive input design
minc,X V(c,X,?) s.t. M(c,X,?) 0

• True system parameter ? is unknown
• Robust designs ? 2 ?
• (Rojas, Welsh, Goodwin and Feuer Automatica 07
  ECC07, Hjalmarsson, Mårtensson and Wahlberg SYSID
  06, Mårtensson and Hjalmarsson SYSID 06)
• Adaptive designs
• Certainty equivalence approach
• On-line parameter estimation
• On-line input design adaptation
• (Lindqvist and Hjalmarsson CDC01, Gerenscér and
  Hjalmarsson CDC05)

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Adaptive input design
Adaptive
Adaptive control but with non-standard cost
function!
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Insights

• Ensure persistence of excitation )
• Parameter convergence at suff. rate )
• Filter quickly close to optimal )
• Off-line accuracy

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A result by Lai and Wei

• Rn?k1n ?k?kT
• ?min (Rn) ! 1
• log ?max (Rn) o( ?min(Rn))
• )

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Parameter convergence
Result (?n-?)O(log(n)/n) w.p.1

• Conditions
• Stable ARX-system A(q)ynq-1B(q)unen
• Least squares estimation in regression model
  yn?nT?en
• Input generated by an m-th order FIR-filter
  unFn(q)wn? fn,i wn-i

) the cis can be taken as auto-correlations of
input

• 0lt? fn,0 8 n 0
• 1gtKf fn,i 8 n 0, i1,...,m (w.p. 1)

(simple to ensure by adding 0ltcmin c0 cmaxlt1
constraints)
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Asymptotic distribution
Result n1/2(?n-?) N(0,P) where P is the
asymptotic covariance matrix of the least squares
estimate when unF(q,?)wn
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Extension to binary signals
unsgn(? fn,j wn-j)

• Result
• Convergence
• Asymptotic distribution
• same as before

• Conditions
• As before but conditions on fn,j relaxed to
• fn,0? 0 8 n 0 w.p. 1

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Numerical illustration L2-gain estimation

• Estimation of L2-system gain G221/(2?) s
  G(ej?)2d?
• Input design problem
• min E un2
• s.t. Var(G(?n)22) ?

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Numerical illustration Results
Parameter convergence
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Numerical illustration Results
Variance of L_2-gain estimate as function of
sample size
Solid line Variance of adaptive scheme (Monte
Carlo simulations) Dotted line Theoretical
value of asymptotic variance using optimal filter
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Summary

• Adaptive approach to handle unknown system
  description
• Certainty equivalence principle
• ARX system / FIR input filter
• Parameter convergence
• Asymptotically, in the sample size, the same
  accuracy as the optimal off-line approach where
  knowledge of the true system is used.

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Thank you for your attention!