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Bounded-Error Identification of Linear Systems with Input and Output Backlash V. Cerone 1 , D. Piga 2 , D. Regruto 1 1 Dipartimento di Automatica e Informatica, Politecnico di Torino, Italy 2 Delft Center for Systems and Control, Delft University of Technology, The Netherlands 16th IFAC Symposium on System Identification Session WeB01: “Block Oriented Nonlinear Identification 2” Brussels, Belgium — July 11, 2012

Bounded-Error Identification of Linear Systems with Input and Output Backlash Dario Piga System Description Linear dynamical system with input backlash η t u t x t w t y t + + ❄ ✲ ✲ ✲ ✲ B L ♠ u t : known input signal y t : noise-corrupted measurement of w t x t : not measurable inner signal Linear dynamical system with output backlash B : backlash nonlinearity η t u t x t w t y t L : linear dynamic subsystem + + ❄ ✲ ✲ ✲ ✲ L B ♠ | η t | ≤ ∆ η t ; ∆ η t known (Set-Membership characterization) DCSC - Delft Center for Systems and Control 1

Bounded-Error Identification of Linear Systems with Input and Output Backlash Dario Piga System Description η t u t x t w t y t ❄ ✲ ✲ ✲ ✲ L B ✐ w t ✻ ✡ ✡ ✡ ✡ ✒ � � m l m r ✡ ✡ ✡ ✡ � ✛ ✲ ✡ ✡ ✡ ✡ � ✢ ✡ ✡ ✡ ✡ � for x t ≤ w t − 1  ✡ ✡ ✡ � m l ( x t + c l ) − c l � ✒ ✡ ✡ ✡  � �  m l ✡ ✡ ✡  − c l � �  for x t ≥ w t − 1 ✡ ✢ ✡ ✡  � � ✲ m r ( x t − c r ) + c r w t = B ( x t ) = ✡ ✡ s s x t � � c r ✒ � ✡ ✡ m r � � � for w t − 1 − c l < x t < w t − 1 ✡ ✡ � � �   ✡ ✡ + c r w t − 1 � � �  ✡ ✢  ✡ � � �  m l m r ✛ ✲ ✒ � ✡ � � � � ✡ � � � � ✡ � � � � ✡ ✢ ✡ � � � � na nb � � L : x t = − a i x t − i + b j u t − j i =1 j =0 DCSC - Delft Center for Systems and Control 2

Bounded-Error Identification of Linear Systems with Input and Output Backlash Dario Piga Identification of linear systems with backlash • Aim of the work: compute bounds on the backlash parameters γ T = [ m l c l m r m r ] and linear block parameters θ T = [ a 1 ... a na b 0 b 1 ... b nb ] . • Parameter bound computation of linear systems with backlash is NP-hard in the size of the experimental data sequence ⇓ Computationally tractable relaxations are needed DCSC - Delft Center for Systems and Control 3

Bounded-Error Identification of Linear Systems with Input and Output Backlash Dario Piga Feasible parameter set (FPS) In bounded-error (or set-membership) context, all the system parameters γ and θ consistent with the measurement data sequence, the assumed model structure and the error bounds are feasible solution to the identification problem (and are said to belong to the feasible parameter set D γθ ). How to construct the Feasible Parameter Set? DCSC - Delft Center for Systems and Control 4

Bounded-Error Identification of Linear Systems with Input and Output Backlash Dario Piga Backlash nonlinearity w t ✻ ✡ ✡ ✡ ✡ m l � ✒ � m r ✡ ✡ ✡ ✡ ✛ ✲ � ✡ ✡ ✡ ✡ ✢ ✡ � ✡ ✡ ✡ � � ✒ ✡ ✡ ✡ � � ✡ ✡ ✡ − c l � � ✢ ✡ ✡ ✡ ✲ � � ✡ ✡ s c r s x t � � � ✒ ✡ ✡ � � � ✡ ✡ � � � ✡ ✡ � � � ✢ ✡ ✡ � � � ✛ ✲ ✒ � ✡ � � � � ✡ � � � � ✡ ✢ ✡ � � � � Can the backlash nonlinearity be inverted? DCSC - Delft Center for Systems and Control 5

Bounded-Error Identification of Linear Systems with Input and Output Backlash Dario Piga Backlash nonlinearity w t ✻ ✡ ✡ ✡ ✡ m l � ✒ � m r ✡ ✡ ✡ ✡ ✛ ✲ � ✡ ✡ ✡ ✡ ✡ ✢ � ✡ ✡ ✡ � ✒ � ✡ ✡ ✡ � � ✡ ✡ ✡ − c l � � ✢ ✡ ✡ ✡ ✲ � � ✡ ✡ s c r s x t � � � ✒ ✡ ✡ � � � ✡ ✡ � � � ✡ ✡ � � � ✢ ✡ ✡ � � � ✛ ✲ � ✒ ✡ � � � � ✡ � � � � ✡ ✡ ✢ � � � � Definition 1: Y r ( right-invertible output sequence ) Y r = { y t ∈ R : y t − y t − 1 > ∆ η t + ∆ η t − 1 } Definition 2: Y l ( left-invertible output sequence ) Y l = { y t ∈ R : y t − y t − 1 < − ∆ η t − ∆ η t − 1 } DCSC - Delft Center for Systems and Control 6

Bounded-Error Identification of Linear Systems with Input and Output Backlash Dario Piga Backlash nonlinearity w t ✻ ✡ ✡ ✡ ✡ m l ✒ � � m r ✡ ✡ ✡ ✡ ✛ ✲ � ✡ ✡ ✡ ✡ ✢ ✡ � ✡ ✡ ✡ � ✡ ✡ ✡ ✒ � � � ✡ ✡ ✡ − c l � � ✡ ✢ ✡ ✡ � � ✲ ✡ ✡ c r x t s s � � � ✒ ✡ ✡ � � � ✡ ✡ � � � ✡ ✡ ✢ ✡ � � � ✡ ✛ ✲ � � � ✒ � ✡ � � � � ✡ � � � � ✡ ✢ ✡ � � � � Proposition 1: If y t ∈ Y r ⇒ x t = w t m r + c r Proposition 2: If y t ∈ Y l ⇒ x t = w t m l − c l � w t Proposition 3: If y t ∈ Y r ∪ Y l ⇒ x t = � � w t � m r + c r χ Y r ( y t ) + m l − c l χ Y l ( y t ) ⇒ m r m l x k = m l ( y k − η k + m r c r ) χ Y r ( y k )+ m r ( y k − η k − m l c l ) χ Y l ( y k ) DCSC - Delft Center for Systems and Control 7

Bounded-Error Identification of Linear Systems with Input and Output Backlash Dario Piga Feasible parameter set (FPS) The FPS D γθ is the projection on the parameter space of the set D of all system parame- ters γ - θ , noise samples and η t and inner signals x t consistent with the measurement data sequence, the assumed model structure and the error bounds, given by: na nb � � � D = ( γ, θ, x, η ) : x k = − a i x k − i + b j u k − j ; i =1 j =1 m r m l x k = m l ( y k − η k + m r c r ) χ Y r ( y k ) + m r ( y k − η k − m l c l ) χ Y l ( y k ); � | η k | ≤ ∆ η k , k : y k ∈ Y r ∪ Y l DCSC - Delft Center for Systems and Control 8

Bounded-Error Identification of Linear Systems with Input and Output Backlash Dario Piga Computation of parameter bounds • Exact parameter bounds: γ k = min γ k = max ( γ,θ,x,η ) ∈D γ k , ( γ,θ,x,η ) ∈D γ k θ j = min θ j = max ( γ,θ,x,η ) ∈D θ j , ( γ,θ,x,η ) ∈D θ j • Parameter Uncertainty Intervals: � � � � PUI γ k = γ k ; γ k PUI θ j = θ j ; θ j Remark 1: The system parameters γ - θ , the inner signals x t and the noise samples η t are decision variables in the above optimization problem ⇒ The number of optimization vari- ables increases with the number of measurements Remark 2: D is a nonconvex set described by polynomial constraints ⇒ exact bound com- putation requires to solve a set of nonconvex optimization problems DCSC - Delft Center for Systems and Control 9

Bounded-Error Identification of Linear Systems with Input and Output Backlash Dario Piga Computation of parameter bounds • Standard nonlinear optimization tools can not be exploited to compute bounds on γ k (resp. θ j ) since they can trap in local minima ⇓ The true value is not guaranteed to lie within the computed bounds • Relax original identification problems to convex optimization problems ⇓ Guaranteed (relaxed) bounds on each parameter γ k (resp. θ j ) can be evaluated DCSC - Delft Center for Systems and Control 10

Bounded-Error Identification of Linear Systems with Input and Output Backlash Dario Piga Computation of relaxed PUI : LMI relaxation • General Idea Exploit LMI relaxation for semialgebraic optimization problems SOS decomposition ( G. Chesi et al. (1999), P. Parrillo (2003) ) Theory of moments ( J. B. Lasserre (2001) ) • Computational complexity Due to the large number of optimization variables and constraints involved in the identi- fication problems, such LMI relaxation techniques leads, in general, to untractable SDP problems The peculiar structured sparsity of the formulated identification problems can be used to reduce the computational complexity of such LMI-relaxation techniques in computing parameter bounds DCSC - Delft Center for Systems and Control 11

Bounded-Error Identification of Linear Systems with Input and Output Backlash Dario Piga Computation of relaxed bounds: exploiting sparsity na nb � � � D = ( γ, θ, x, η ) : x k = − a i x k − i + b j u k − j ; i =1 j =1 m r m l x k = m l ( y k − η k + m r c r ) χ Y r ( y k ) + m r ( y k − η k − m l c l ) χ Y l ( y k ); � | η k | ≤ ∆ η k , k : y k ∈ Y r ∪ Y l na nb � � • x k = − a i x k − i + b j u k − j only depends on the linear system parameters a i i =1 j =1 and b j and on the inner signal samples x k , . . . , x k − na • m r m l x k = m l ( y k − η k + m r c r ) χ Y r ( y k ) + m r ( y k − η k − m l c l ) χ Y l ( y k ) only de- pends on the backlash parameters m l , c l , m r , c r and on noise sample η k • | η k | ≤ ∆ η k only depends on the noise sample η k DCSC - Delft Center for Systems and Control 12