Title: Adaptive input design for ARX systems
1Adaptive input design for ARX systems
László Gerencsér MTA SZTAKI Hungarian Academy of
Science Budapest, Hugary Håkan
Hjalmarsson School of Electrical
Engineering KTH Stockholm, Sweden
2Input Design for Identification
Leads to optimization problems that are
typically non-convex and infinite-dimensional,
and dependent on true system
3Outline
- 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
4Prediction 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
5A Typical Input Design Problem
- Quantity of interest J(G0)
Examples Static gain G0(1),
Impulse response coefficient,
Zero, Pole.
6Mathematical Formulation
7Convexification
Schur complement)
Convex in ?ubut infinite dimensional problem
8Finite dimensionalization
But now infinite dimensional constraint in
?! KYP) LMI constraint
9Input 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
10 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
11Adaptive 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)
12Adaptive input design
Adaptive
Adaptive control but with non-standard cost
function!
13Insights
- Ensure persistence of excitation )
- Parameter convergence at suff. rate )
- Filter quickly close to optimal )
- Off-line accuracy
14A result by Lai and Wei
- Rn?k1n ?k?kT
- ?min (Rn) ! 1
- log ?max (Rn) o( ?min(Rn))
- )
15Parameter 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)
16Asymptotic distribution
Result n1/2(?n-?) N(0,P) where P is the
asymptotic covariance matrix of the least squares
estimate when unF(q,?)wn
17Extension 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
18Numerical 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) ?
19Numerical illustration Results
Parameter convergence
20Numerical 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
21Summary
- 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.
22Thank you for your attention!