Identification of Input/Output LPV Models V. Laurain, M. Gilson, R. - PowerPoint PPT Presentation

Identification of Input/Output LPV Models V. Laurain, M. Gilson, R. T´ oth, H. Garnier CRAN, Nancy-Universit´ e, DCSC, TU Delft Delft Center for System Delft Center for System s and Control s and Control V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 1 / 32

Outline Introduction 1 Problem Description 2 LPV model description Scheduling dependency model Prediction Error for LPV systems 3 Usual linear regression The literature methods Reformulation of the problem RIV Algorithm Results 4 LPV-RIV performance Application example 5 Conclusion 6 V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 2 / 32

Introduction Outline Introduction 1 2 Problem Description LPV model description Scheduling dependency model Prediction Error for LPV systems 3 Usual linear regression The literature methods Reformulation of the problem RIV Algorithm Results 4 LPV-RIV performance 5 Application example Conclusion 6 V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 3 / 32

Introduction What is an LPV model ? Definition Linear relationship between the input and output The model parameters vary in time according to an external variable : The scheduling variable Identification of LPV models Control theory of LPV models well developed (Gain scheduling) LPV State-Space identification methods : Multiple models approaches, LMI’s based methods, Gradient methods Input/Output identification methods : Linear regression methods , Nonlinear optimization Advantages Tradeoff between LTI systems and NL systems Wide range of behavior representation capability Many practical applications (environment, automotive, wafer scanners...) V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 4 / 32

Introduction What is an LPV model ? Definition Linear relationship between the input and output The model parameters vary in time according to an external variable : The scheduling variable Identification of LPV models Control theory of LPV models well developed (Gain scheduling) LPV State-Space identification methods : Multiple models approaches, LMI’s based methods, Gradient methods Input/Output identification methods : Linear regression methods , Nonlinear optimization Advantages Tradeoff between LTI systems and NL systems Wide range of behavior representation capability Many practical applications (environment, automotive, wafer scanners...) V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 4 / 32

Introduction What is an LPV model ? Definition Linear relationship between the input and output The model parameters vary in time according to an external variable : The scheduling variable Identification of LPV models Control theory of LPV models well developed (Gain scheduling) LPV State-Space identification methods : Multiple models approaches, LMI’s based methods, Gradient methods Input/Output identification methods : Linear regression methods , Nonlinear optimization Advantages Tradeoff between LTI systems and NL systems Wide range of behavior representation capability Many practical applications (environment, automotive, wafer scanners...) V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 4 / 32

Introduction LPV modeling concept of a plane Linear system for a given altitude The system parameters vary with the altitude V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 5 / 32

Introduction LPV modeling concept of a bar under constraint Consider a metal bar under a constraint F . F x The displacement is x For a constant temperature : x = kF with k the stiffness coefficient When considering the temperature θ , the LPV model can be written : x = k ( θ ) F V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 6 / 32

Problem Description Outline Introduction 1 2 Problem Description LPV model description Scheduling dependency model Prediction Error for LPV systems 3 Usual linear regression The literature methods Reformulation of the problem RIV Algorithm Results 4 LPV-RIV performance 5 Application example Conclusion 6 V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 7 / 32

Problem Description LPV model description System LPV IO model LTI ( A ( q − 1 ) χ ( t k ) = B ( q − 1 ) u ( t k ) M y ( t k ) = χ ( t k ) + v ( t k ) χ is the noise-free output u is the input v is the additive noise with bounded spectral density y is the noisy output of the system q is the time-shift operator, i.e. q − i u ( t k ) = u ( t k − i ) V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 8 / 32

Problem Description LPV model description System LPV IO model LPV ( A ( p k , q − 1 ) χ ( t k ) = B ( p k , q − 1 ) u ( t k ) M y ( t k ) = χ ( t k ) + v ( t k ) χ is the noise-free output u is the input v is the additive noise with bounded spectral density y is the noisy output of the system q is the time-shift operator, i.e. q − i u ( t k ) = u ( t k − i ) p is the scheduling parameter V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 8 / 32

Problem Description LPV model description System LPV IO model LTI ( A ( q − 1 ) χ ( t k ) = B ( q − 1 ) u ( t k ) M y ( t k ) = χ ( t k ) + v ( t k ) A ( q − 1 ) and B ( q − 1 ) are polynomials in q − 1 of degree n a and n b respectively : n a X A ( q − 1 ) = 1 + a i q − i i =1 n b X B ( q − 1 ) = b j q − i , j =0 V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 8 / 32

Problem Description LPV model description System LPV IO model LPV ( A ( p k , q − 1 ) χ ( t k ) = B ( p k , q − 1 ) u ( t k ) M y ( t k ) = χ ( t k ) + v ( t k ) A ( p k , q − 1 ) and B ( p k , q − 1 ) are polynomials in q − 1 of degree n a and n b respectively : n a X A ( p k , q − 1 ) = 1 + a i ( p k ) q − i i =1 n b X B ( p k , q − 1 ) = b j ( p k ) q − i , j =0 V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 8 / 32

Problem Description Scheduling dependency model Model Process Model “ ” A ( p k , q − 1 , ρ ) , B ( p k , q − 1 , ρ ) ` ´ G ρ : = A ρ , B ρ where the p dependent polynomials A and B are parameterized as n a A ρ : A ( p k , q − 1 , ρ ) = 1 + a i ( p k ) q − i , X i =1 n b B ρ : B ( p k , q − 1 , ρ ) = b j ( p k ) q − i . X j =0 Usual assumption : static dependency on p n α X a i ( p k ) = a i , 0 + a i , l f l ( p k ) i = 1 , . . . , n a l =1 n β X b j ( p k ) = b j , 0 + b j , l g l ( p k ) j = 0 , . . . , n b l =1 n β { f l } n α l =1 and { g l } l =1 are analytic functions of p known a priori (polynomials for example) V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 9 / 32

Problem Description Scheduling dependency model Model Process Model “ ” A ( p k , q − 1 , ρ ) , B ( p k , q − 1 , ρ ) ` ´ G ρ : = A ρ , B ρ where the p dependent polynomials A and B are parameterized as n a A ρ : A ( p k , q − 1 , ρ ) = 1 + a i ( p k ) q − i , X i =1 n b B ρ : B ( p k , q − 1 , ρ ) = b j ( p k ) q − i . X j =0 Usual assumption : static dependency on p n α X a i ( p k ) = a i , 0 + a i , l f l ( p k ) i = 1 , . . . , n a l =1 n β X b j ( p k ) = b j , 0 + b j , l g l ( p k ) j = 0 , . . . , n b l =1 Parameter vector ρ = [ a 1 , 0 . . . a n a , n α , b 0 , 0 . . . b n b , n β ] ⊤ V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 9 / 32

Prediction Error for LPV systems Outline Introduction 1 2 Problem Description LPV model description Scheduling dependency model Prediction Error for LPV systems 3 Usual linear regression The literature methods Reformulation of the problem RIV Algorithm Results 4 LPV-RIV performance 5 Application example Conclusion 6 V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 10 / 32

Prediction Error for LPV systems Usual linear regression ( A ( p k , q − 1 , ρ ) χ ( t k )= B ( p k , q − 1 , ρ ) u ( t k ) M θ y ( t k )= χ ( t k ) + v ( t k ) V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 11 / 32

Prediction Error for LPV systems Usual linear regression ( A ( p k , q − 1 , ρ ) χ ( t k )= B ( p k , q − 1 , ρ ) u ( t k ) M θ A ( p k , q − 1 , ρ ) y ( t k )= A ( p k , q − 1 , ρ ) χ ( t k ) + A ( p k , q − 1 , ρ ) v ( t k ) A ( p k , q − 1 , ρ ) y ( t k )= B ( p k , q − 1 , ρ ) u ( t k ) + A ( p k , q − 1 , ρ ) v ( t k ) V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 11 / 32

Prediction Error for LPV systems Usual linear regression A ( p k , q − 1 , ρ ) y ( t k )= B ( p k , q − 1 , ρ ) u ( t k ) + A ( p k , q − 1 , ρ ) v ( t k ) ! n b ! n a X X a i ( p k ) q − i b j ( p k ) q − j u ( t k ) + A ( p k , q − 1 , ρ ) v ( t k ) 1 + y ( t k )= i =1 j =0 V. Laurain, M. Gilson, R. T´ oth, H. Garnier LPV-RIV () S´ eminaire LOUVAIN 11 / 32