Optimal local LPV identification experiment design D. Ghosh (1) , X. Bombois (1) , J. Huillery (1) , G. Scorletti (1) et ere (2) G. Merc` 1. Laboratoire Amp` ere UMR CNRS 5005 2. LIAS, Universit´ e de Poitiers ERNSI workshop - 28 September 2016 X. Bombois (CNRS) OED-LPV 28/09/2016 1 / 21
Introduction An LPV system is a system whose parameters depend on an exogenous (scheduling) variable p ( t ) If p ( t ) is kept constant , the LPV system is an LTI system The dynamics of this LTI system depend on the value of the constant p We have a collection of LTI dynamics at different operating points Such a representation can be used to deal with non-linear systems (gain scheduling) X. Bombois (CNRS) OED-LPV 28/09/2016 2 / 21
Introduction An LPV system is a system whose parameters depend on an exogenous (scheduling) variable p ( t ) If p ( t ) is kept constant , the LPV system is an LTI system The dynamics of this LTI system depend on the value of the constant p We have a collection of LTI dynamics at different operating points Such a representation can be used to deal with non-linear systems (gain scheduling) X. Bombois (CNRS) OED-LPV 28/09/2016 2 / 21
Introduction An LPV system is a system whose parameters depend on an exogenous (scheduling) variable p ( t ) If p ( t ) is kept constant , the LPV system is an LTI system The dynamics of this LTI system depend on the value of the constant p We have a collection of LTI dynamics at different operating points Such a representation can be used to deal with non-linear systems (gain scheduling) X. Bombois (CNRS) OED-LPV 28/09/2016 2 / 21
Introduction Local LPV identification approach: p ( t ) is kept constant at successive operating points and local LTI identification experiments are performed We determine those operating points and the local LTI identification experiments to guarantee a certain model accuracy with the least input energy Related work on the selection of the scheduling sequence: Khalate et al: 2009 , Vizer et al: 2015 X. Bombois (CNRS) OED-LPV 28/09/2016 3 / 21
Introduction Local LPV identification approach: p ( t ) is kept constant at successive operating points and local LTI identification experiments are performed We determine those operating points and the local LTI identification experiments to guarantee a certain model accuracy with the least input energy Related work on the selection of the scheduling sequence: Khalate et al: 2009 , Vizer et al: 2015 X. Bombois (CNRS) OED-LPV 28/09/2016 3 / 21
Introduction Local LPV identification approach: p ( t ) is kept constant at successive operating points and local LTI identification experiments are performed We determine those operating points and the local LTI identification experiments to guarantee a certain model accuracy with the least input energy Related work on the selection of the scheduling sequence: Khalate et al: 2009 , Vizer et al: 2015 X. Bombois (CNRS) OED-LPV 28/09/2016 3 / 21
Description of the LPV system We consider the following LPV-OE system for simplicity: n a n b � � a 0 b 0 y nf ( t ) = − i ( p ( t )) y ( t − i ) + i ( p ( t )) u ( t − i ) i =1 i =1 y ( t ) = y nf ( t ) + e ( t ) 1 ( p ( t )) , ...., b n b ( p ( t ))) T depends on The parameter vector ξ 0 ( p ( t )) = ( a 0 the time-varying scheduling variable p ( t ) n p � a 0 i ( p ( t )) = a 0 a 0 i , j p j ( t ) i , 0 + j =1 n p � b 0 i ( p ( t )) = b 0 b 0 i , j p j ( t ) i , 0 + j =1 X. Bombois (CNRS) OED-LPV 28/09/2016 4 / 21
Description of the LPV system n p � a 0 i ( p ( t )) = a 0 a 0 i , j p j ( t ) i , 0 + i = 1 ... n a j =1 n p � b 0 i ( p ( t )) = b 0 b 0 i , j p j ( t ) i , 0 + i = 1 ... n b j =1 This defines a mapping T ( p ( t )) between the global parameter vector θ 0 and the time-varying parameter vector ξ 0 ( p ( t )) ξ 0 ( p ( t )) = T ( p ( t )) θ 0 θ 0 = ( a 0 ξ 0 ( p ( t )) = ( a 0 1 ( p ( t )) , ...., b n b ( p ( t ))) T 1 , 0 , ..., b 0 n b , n p ) T X. Bombois (CNRS) OED-LPV 28/09/2016 5 / 21
Identification objective ξ 0 ( p ( t )) = T ( p ( t )) θ 0 The parameter vector θ 0 entirely determines the LPV system Objective. Determine with the least powerful excitation an estimate ˆ θ of θ 0 having a given accuracy: P − 1 > R adm θ X. Bombois (CNRS) OED-LPV 28/09/2016 6 / 21
Identification objective ξ 0 ( p ( t )) = T ( p ( t )) θ 0 The parameter vector θ 0 entirely determines the LPV system Objective. Determine with the least powerful excitation an estimate ˆ θ of θ 0 having a given accuracy: P − 1 > R adm θ X. Bombois (CNRS) OED-LPV 28/09/2016 6 / 21
Identification of an LPV system: local approach Suppose p ( t ) is kept constant to an operating point p m p ( t ) = p m ∀ t The LPV system then reduces to an LTI system described by a time-invariant parameter vector ξ 0 ( p m ) y ( t ) = G ( z , ξ 0 ( p m )) u ( t ) + e ( t ) ξ 0 ( p m ) = T ( p m ) θ 0 This LTI system can of course then be identified using LTI prediction error identification X. Bombois (CNRS) OED-LPV 28/09/2016 7 / 21
Identification of an LPV system: local approach Suppose p ( t ) is kept constant to an operating point p m p ( t ) = p m ∀ t The LPV system then reduces to an LTI system described by a time-invariant parameter vector ξ 0 ( p m ) y ( t ) = G ( z , ξ 0 ( p m )) u ( t ) + e ( t ) ξ 0 ( p m ) = T ( p m ) θ 0 This LTI system can of course then be identified using LTI prediction error identification X. Bombois (CNRS) OED-LPV 28/09/2016 7 / 21
LTI identification at an operating point p m If we apply an input signal u m of spectrum Φ u m to y m ( t ) = G ( z , ξ 0 ( p m )) u m ( t ) + e m ( t ) , we can collect a data set Z N m = { y m ( t ) , u m ( t ) | t = 1 ... N } and identify an estimate ˆ ξ m of ξ 0 ( p m ) using: N 1 � ˆ ( y m ( t ) − G ( z , ξ ) u m ( t )) 2 ξ m = arg min N ξ t =1 This estimate is (asymptotically) such that ˆ ξ m ∼ N ( ξ 0 ( p m ) , P ˆ ξ m ) X. Bombois (CNRS) OED-LPV 28/09/2016 8 / 21
LTI identification at an operating point p m The estimate ˆ ξ m is (asymptotically) such that ˆ ξ m ∼ N ( ξ 0 ( p m ) , P ˆ ξ m ) ξ m depends on ξ 0 ( p m ) and Φ u m : The covariance matrix P ˆ � π ξ m = N 1 F ( e j ω , ξ 0 ( p m )) F ( e j ω , ξ 0 ( p m )) ∗ Φ u m ( ω ) d ω P − 1 ˆ σ 2 2 π − π e � F ( z , ξ 0 ( p m )) = dG ( z , ξ ) � � d ξ � ξ 0 ( p m ) This operation has to be repeated at different p m to deduce an estimate of θ 0 since dim ( ξ 0 ( p m )) < dim ( θ 0 ) X. Bombois (CNRS) OED-LPV 28/09/2016 9 / 21
LTI identification at an operating point p m The estimate ˆ ξ m is (asymptotically) such that ˆ ξ m ∼ N ( ξ 0 ( p m ) , P ˆ ξ m ) ξ m depends on ξ 0 ( p m ) and Φ u m : The covariance matrix P ˆ � π ξ m = N 1 F ( e j ω , ξ 0 ( p m )) F ( e j ω , ξ 0 ( p m )) ∗ Φ u m ( ω ) d ω P − 1 ˆ σ 2 2 π − π e � F ( z , ξ 0 ( p m )) = dG ( z , ξ ) � � d ξ � ξ 0 ( p m ) This operation has to be repeated at different p m to deduce an estimate of θ 0 since dim ( ξ 0 ( p m )) < dim ( θ 0 ) X. Bombois (CNRS) OED-LPV 28/09/2016 9 / 21
ξ m of ξ 0 ( p m ) = T ( p m ) θ 0 : We obtain M estimates ˆ ξ m = T ( p m ) θ 0 + δ m ˆ δ m ∼ N (0 , P ˆ ξ m ) θ of θ 0 is classically determined using ordinary least squares The estimate ˆ based on the observations ˆ ξ m and the regressor T ( p m ) This is however not the minimum variance estimator since the respective variances of ˆ ξ m are neglected = ⇒ use of weighted least squares: M � � T � � � ˆ ˆ P − 1 ˆ ξ m − T ( p m ) θ ξ m − T ( p m ) θ θ = arg min ˆ ξ m θ m =1 X. Bombois (CNRS) OED-LPV 28/09/2016 10 / 21
ξ m of ξ 0 ( p m ) = T ( p m ) θ 0 : We obtain M estimates ˆ ξ m = T ( p m ) θ 0 + δ m ˆ δ m ∼ N (0 , P ˆ ξ m ) θ of θ 0 is classically determined using ordinary least squares The estimate ˆ based on the observations ˆ ξ m and the regressor T ( p m ) This is however not the minimum variance estimator since the respective variances of ˆ ξ m are neglected = ⇒ use of weighted least squares: M � � T � � � ˆ ˆ P − 1 ˆ ξ m − T ( p m ) θ ξ m − T ( p m ) θ θ = arg min ˆ ξ m θ m =1 X. Bombois (CNRS) OED-LPV 28/09/2016 10 / 21
ξ m of ξ 0 ( p m ) = T ( p m ) θ 0 : We obtain M estimates ˆ ξ m = T ( p m ) θ 0 + δ m ˆ δ m ∼ N (0 , P ˆ ξ m ) θ of θ 0 is classically determined using ordinary least squares The estimate ˆ based on the observations ˆ ξ m and the regressor T ( p m ) This is however not the minimum variance estimator since the respective variances of ˆ ξ m are neglected = ⇒ use of weighted least squares: M � � T � � � ˆ ˆ P − 1 ˆ ξ m − T ( p m ) θ ξ m − T ( p m ) θ θ = arg min ˆ ξ m θ m =1 X. Bombois (CNRS) OED-LPV 28/09/2016 10 / 21
M � � T � � � ˆ ˆ P − 1 ˆ ξ m − T ( p m ) θ ξ m − T ( p m ) θ θ = arg min ˆ ξ m θ m =1 The estimate ˆ θ is such that ˆ θ ∼ N ( θ 0 , P θ ) M � P − 1 T T ( p m ) P − 1 = ξ m T ( p m ) ˆ θ m =1 with P − 1 ξ m linear in Φ u m ˆ P − 1 is the sum of the contribution of each local experiments !! θ X. Bombois (CNRS) OED-LPV 28/09/2016 11 / 21
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