Background Problem Formulation Subspace-based algorithm Main Result Examples Conclusions A SUBSPACE-BASED METHOD FOR SOLVING LAGRANGE-SYLVESTER INTERPOLATION PROBLEMS Hüseyin Akçay Department of Electrical and Electronics Engineering Anadolu University, Eski¸ sehir, Turkey September 29, 2008 Hüseyin Akçay A SUBSPACE-BASED METHOD FOR SOLVING LAGRANGE-SYL
Background Problem Formulation Subspace-based algorithm Main Result Examples Conclusions Outline Background 1 Problem Formulation 2 Subspace-based algorithm 3 Derivation of the algorithm Projection onto the observability range space Extracting A and C matrices Extracting B and D matrices from data Summary of the subspace-based interpolation algorithm Main Result 4 Comparison of the algorithm with existing methods Examples 5 Conclusions 6 Hüseyin Akçay A SUBSPACE-BASED METHOD FOR SOLVING LAGRANGE-SYL
Background Problem Formulation Subspace-based algorithm Main Result Examples Conclusions Interpolation of matrix valued rational functions analytic at infinity from frequency domain data ... Lagrange interpolation studied by Antoulas and Anderson 1 using a tool called Löwner matrix also with additional constraints such as bounded real, positive real etc. Generating system approach studied by Antoulas, Ball, 2 Kang, Willems, Gohberg, and Rodman. Applications of interpolation theory to control and system 3 theory and estimation (see, for example, the monographs: Ball, Gohberg, and Rodman; Nikolski). Hüseyin Akçay A SUBSPACE-BASED METHOD FOR SOLVING LAGRANGE-SYL
Background Problem Formulation Subspace-based algorithm Main Result Examples Conclusions Consider a multi-input/multi-output, linear-time invariant discrete-time system represented by the state-space equations: x ( t + 1 ) = Ax ( t ) + Bu ( t ) y ( t ) = Cx ( t ) + Du ( t ) where x ( t ) ∈ R n is the state, u ( t ) ∈ R m and y ( t ) ∈ R p are the input and the output. Transfer function G ( z ) = D + C ( zI n − A ) − 1 B is stable and { A , B } and { A , C } are controllable and observable. Hüseyin Akçay A SUBSPACE-BASED METHOD FOR SOLVING LAGRANGE-SYL
Background Problem Formulation Subspace-based algorithm Main Result Examples Conclusions Given: samples of G ( z ) and its derivatives at L distinct points z k ∈ D d j G ( z k ) = w kj , j = 0 , 1 , · · · , N k ; k = 1 , 2 , · · · , L . dz j B , � Find: ( � A , � C , � D ) , a minimal realization of G ( z ) . Lagrange-Sylvester rational interpolation problem. • Obvious solution! Reduce the problem first to a system of independent scalar problems and obtain a minimal solution by eliminating unobservable or/and uncontrollable modes. Hüseyin Akçay A SUBSPACE-BASED METHOD FOR SOLVING LAGRANGE-SYL
Background Problem Formulation Subspace-based algorithm Main Result Examples Conclusions • (Bi)tangential and contour integral versions treated for example, in Ball, Gohberg, and Rodman. • Related problems: Nonhomogeneous interpolation with metric constraints; Nevanlinna-Pick interpolation; Partial realization. Hüseyin Akçay A SUBSPACE-BASED METHOD FOR SOLVING LAGRANGE-SYL
Background Derivation of the algorithm Problem Formulation Projection onto the observability range space Subspace-based algorithm Extracting A and C matrices Main Result Extracting B and D matrices from data Examples Summary of the subspace-based interpolation algorithm Conclusions Outline Background 1 Problem Formulation 2 Subspace-based algorithm 3 Derivation of the algorithm Projection onto the observability range space Extracting A and C matrices Extracting B and D matrices from data Summary of the subspace-based interpolation algorithm Main Result 4 Comparison of the algorithm with existing methods Examples 5 Conclusions 6 Hüseyin Akçay A SUBSPACE-BASED METHOD FOR SOLVING LAGRANGE-SYL
Background Derivation of the algorithm Problem Formulation Projection onto the observability range space Subspace-based algorithm Extracting A and C matrices Main Result Extracting B and D matrices from data Examples Summary of the subspace-based interpolation algorithm Conclusions Take the z -transform of the state-space equations: zX ( z ) = AX ( z ) + BU ( z ) Y ( z ) = CX ( z ) + DU ( z ) where X ( z ) denotes the z -transforms of x ( k ) defined by ∞ � U ( z ) ∆ u ( k ) z − k . = k = 0 Let X j ( z ) be the resulting state z -transform when u ( k ) = e j . Hüseyin Akçay A SUBSPACE-BASED METHOD FOR SOLVING LAGRANGE-SYL
Background Derivation of the algorithm Problem Formulation Projection onto the observability range space Subspace-based algorithm Extracting A and C matrices Main Result Extracting B and D matrices from data Examples Summary of the subspace-based interpolation algorithm Conclusions Define the compound state z -transform matrix: X C ( z ) ∆ = [ X 1 ( z ) X 2 ( z ) · · · X m ( z )] . Then, G ( z ) can implicitly be described as G ( z ) = CX C ( z ) + D with zX C ( z ) = AX C ( z ) + B . By recursive use, we obtain the relation k − 1 � CA k X C ( z ) + Dz k + z k G ( z ) CA k − 1 − j Bz j , = k ≥ 1 . j = 0 Hüseyin Akçay A SUBSPACE-BASED METHOD FOR SOLVING LAGRANGE-SYL
Background Derivation of the algorithm Problem Formulation Projection onto the observability range space Subspace-based algorithm Extracting A and C matrices Main Result Extracting B and D matrices from data Examples Summary of the subspace-based interpolation algorithm Conclusions The impulse response coefficients of G ( z ) : � D , k = 0 ; g k = CA k − 1 B , k ≥ 1 . Thus, k � z k G ( z ) = CA k X C ( z ) + g k − j z j , k ≥ 0 . j = 0 Hüseyin Akçay A SUBSPACE-BASED METHOD FOR SOLVING LAGRANGE-SYL
Background Derivation of the algorithm Problem Formulation Projection onto the observability range space Subspace-based algorithm Extracting A and C matrices Main Result Extracting B and D matrices from data Examples Summary of the subspace-based interpolation algorithm Conclusions Hence, G ( z ) I m zG ( z ) zI m = O q X C ( z ) + Γ q . . . . . . z q − 1 G ( z ) z q − 1 I m where C g 0 0 · · · 0 CA g 1 g 0 · · · 0 ∆ ∆ O q = Γ q = , . . . . . ... . . . . . . . . CA q − 1 · · · g q − 1 g q − 2 g 0 Hüseyin Akçay A SUBSPACE-BASED METHOD FOR SOLVING LAGRANGE-SYL
Background Derivation of the algorithm Problem Formulation Projection onto the observability range space Subspace-based algorithm Extracting A and C matrices Main Result Extracting B and D matrices from data Examples Summary of the subspace-based interpolation algorithm Conclusions O q , extended observability matrix , has full rank n if ( C , A ) is observable and q ≥ n . Let · · · 0 0 1 1 0 z ∆ ∆ ∈ R q × q , 0 1 0 Z q ( z ) = J q , 2 = , . . . . . ... . . . . z q − 1 0 · · · 1 0 J 0 J q , 1 = I q , q , 2 = I q . Hüseyin Akçay A SUBSPACE-BASED METHOD FOR SOLVING LAGRANGE-SYL
Background Derivation of the algorithm Problem Formulation Projection onto the observability range space Subspace-based algorithm Extracting A and C matrices Main Result Extracting B and D matrices from data Examples Summary of the subspace-based interpolation algorithm Conclusions J q , 2 obtained by shifting the elements of J q , 1 one row down and filling its first row with zeros. Let J q , j denote the matrix obtained by j − 1 repeated applications of this process to J q , 1 . Note the following relations � J j − 1 j ≤ q q , 2 , J q , j = 0 , j > q . Hüseyin Akçay A SUBSPACE-BASED METHOD FOR SOLVING LAGRANGE-SYL
Background Derivation of the algorithm Problem Formulation Projection onto the observability range space Subspace-based algorithm Extracting A and C matrices Main Result Extracting B and D matrices from data Examples Summary of the subspace-based interpolation algorithm Conclusions Thus, q − 1 � Γ q = J q , 1 + j ⊗ g j j = 0 A compact expression: q − 1 � [ J j Z q ( z ) ⊗ G ( z ) = O q X C ( z ) + q , 2 ⊗ g j ] [ Z q ( z ) ⊗ I m ] . j = 0 • Forms the basis of the frequency domain subspace identification algorithms (McKelvey, Akçay, and Ljung; 1996). Hüseyin Akçay A SUBSPACE-BASED METHOD FOR SOLVING LAGRANGE-SYL
Background Derivation of the algorithm Problem Formulation Projection onto the observability range space Subspace-based algorithm Extracting A and C matrices Main Result Extracting B and D matrices from data Examples Summary of the subspace-based interpolation algorithm Conclusions (Subspace ID: evaluate this equation at a set of distinct points on the unit circle and stack into columns of long matrices yielding a matrix equation affine in O q . Then, recover the range space of O q by a projection.) Differentiate Z q ( z ) ⊗ G ( z ) l times with respect to z : � l � � l � � H ( l ) Z ( j ) q ( z ) ⊗ G ( l − j ) ( z ) q ( z ) = j j = 0 q − 1 � � d l X C ( z ) � [ J j Z ( l ) = O q + q , 2 ⊗ g j ] q ( z ) ⊗ I m , l ≥ 0 d z k j = 0 Hüseyin Akçay A SUBSPACE-BASED METHOD FOR SOLVING LAGRANGE-SYL
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