Computer algebra methods for testing the structural stability of multidimensional systems Yacine Bouzidi ⋆ , Alban Quadrat ⋆⋆ , Fabrice Rouillier ⋆⋆⋆ ⋆ INRIA Saclay - Île-de-France, Disco project ⋆⋆ INRIA Lille, Nord Europe, NON-A Project ⋆⋆⋆ INRIA Paris - Roquencourt, Ouragan ⋆ yacine.bouzidi@inria.fr , ⋆⋆ alban.quadrat@inria.fr , ⋆⋆⋆ Fabrice.Rouillier@inria.fr supported by the ANR MSDOS Journées Nationales de Calcul Formel, Cluny, 2-6 Novembre 2015
Overview Multidimensional systems 1 Structural stability 2 Contribution on the stability test 3 Ongoing work on the stability analysis 4 2/23 Yacine Bouzidi, Alban Quadrat, Fabrice Rouillier Stability of n -dimensional systems
Overview Multidimensional systems 1 Structural stability 2 Contribution on the stability test 3 Ongoing work on the stability analysis 4 3/23 Yacine Bouzidi, Alban Quadrat, Fabrice Rouillier Stability of n -dimensional systems
Discrete-time linear shift-invariant systems • Let ( y n ) n ∈ N and ( u n ) n ∈ N be 2 sequences satisfying the equation: y n + 2 − 3 y n + 1 + 2 y n = 2 u n + 1 + 2 u n , y 0 = 0 , ( ⋆ ) y 1 = 0 . • Definition: The Z -transform of a sequence = ( x n ) n ∈ Z is defined by: � x n z − n . Z (( x n ) n ∈ Z )( z ) := n ∈ Z 2 ( z − 1 + 1 ) 2 ( z 2 + z ) ( ⋆ ) ⇒ Z ( y )( z ) = z − 2 − 3 z − 1 + 2 Z ( u )( z ) = 2 z 2 − 3 z + 1 Z ( u )( z ) . • Definition: The rational function � s j = 0 b j z − j P ( z ) := � r i = 0 a i z − i is called the transfer function of the discrete-time linear system: r s � � a i y i = b j u j . i = 0 j = 0 4/23 Yacine Bouzidi, Alban Quadrat, Fabrice Rouillier Stability of n -dimensional systems
Discrete multidimensional systems • Roesser model: � � � � x h ( i + 1 , j ) x h ( i , j ) = A + B u ( i , j ) , x v ( i , j + 1 ) x v ( i , j ) � � x h ( i , j ) y ( i , j ) = C + D u ( i , j ) . x v ( i , j ) • Fornasini-Marchesini models: x ( i + 1 , j + 1 ) = A 1 x ( i + 1 , j )+ A 2 x ( i , j + 1 )+ B 1 u ( i + 1 , j )+ B 2 u ( i , j + 1 ) , . . . • n -dimensional recursive filters: digitial image processing, . . . j := ( j 1 , . . . , j n ) , k := ( k 1 , . . . , k n ) ∈ Z n ⇒ k − j := ( k 1 − j 1 , . . . , k n − j n ) z − k := z − k 1 � h ( k ) z − k , . . . z − k n Z (( h ( k )) k ∈ Z n )( z ) := . 1 n k ∈ Z n � y ( k ) = ( h ⋆ u )( k ) := h ( k − j ) u ( j ) ⇒ Z ( y )( z ) = Z ( h )( z ) Z ( u )( z ) , j ∈ Z n Z ( h )( z 1 , . . . , z n ) = n ( z 1 , . . . , z n ) d ( z 1 , . . . , z n ) ∈ R ( z 1 , . . . , z n ) . 5/23 Yacine Bouzidi, Alban Quadrat, Fabrice Rouillier Stability of n -dimensional systems
Overview Multidimensional systems 1 Structural stability 2 Contribution on the stability test 3 Ongoing work on the stability analysis 4 6/23 Yacine Bouzidi, Alban Quadrat, Fabrice Rouillier Stability of n -dimensional systems
Structural stability • P := N ( z 1 ,..., z n ) D ( z 1 ,..., z n ) ∈ R ( z 1 , . . . , z n ) : a transfer function ( gcd ( D , N ) = 1). • The closed unit polydisc of C n : n := { z = ( z 1 , . . . , z n ) ∈ C n | | z i | ≤ 1 , i = 1 , . . . , n , } . D n , i.e.: • Definition: P is structurally stable if D is devoid from zero in D n : D ( z 1 , . . . , z n ) � = 0 . ( 1 ) ∀ z = ( z 1 , . . . , z n ) ∈ D • The affine algebraic set associated to D : V C ( D ) := { z = ( z 1 , . . . , z n ) ∈ C n | D ( z 1 , . . . , z n ) = 0 } . • Condition (1) is equivalent to: n = ∅ . V C ( D ) ∩ D 7/23 Yacine Bouzidi, Alban Quadrat, Fabrice Rouillier Stability of n -dimensional systems
Previous works The case n = 1 • Check that the complex roots of the polynomial D ( z ) = a n z n + a n − 1 z n − 1 + . . . + a 0 do not belong to D := { z ∈ C | | z | ≤ 1 } . • Several algebraic stability criteria (Jury test, Bistritz test, etc), discrete time analogues of the Routh-Hurwitz criterion. • Based on Cauchy index computation: sign variation in some polynomial sequences. • The complexity of a univariate gcd computation. 8/23 Yacine Bouzidi, Alban Quadrat, Fabrice Rouillier Stability of n -dimensional systems
Previous works Bistritz test • Let D ( z ) := a n z n + a n − 1 z n − 1 + . . . + a 0 and D ⋆ ( z ) := z n D ( z − 1 ) . • Compute the sequence of polynomials { T i ( z ) } i = n ,..., 0 , defined by T n ( z ) := D ( z ) + D ⋆ ( z ) , T n − 1 ( z ) := D ( z ) + D ⋆ ( z ) , ( z − 1 ) T i − 1 ( z ) := δ i + 1 ( 1 + z ) T i ( z ) − T i + 1 ( z ) , z where δ i + 1 := T i + 1 ( 0 ) T i ( 0 ) for i = n − 1 , . . . , 1. • Criterion: The system is stable if and only if the sequence is normal and the number of sign variation in { T n ( 1 ) , . . . , T 0 ( 1 ) } is zero. 8/23 Yacine Bouzidi, Alban Quadrat, Fabrice Rouillier Stability of n -dimensional systems
Previous works The case n>1 • First step: Simplification of the initial condition. • [Strintzis,Huang 1977]: D ( 0 , . . . , 0 , z n ) � = 0 , | z n | ≤ 1 , D ( 0 , . . . , 0 , z n − 1 , z n ) � = 0 , | z n − 1 | ≤ 1 , | z n | = 1 , . . . . . . D ( 0 , z 2 , . . . , z n ) � = 0 , | z 2 | ≤ 1 , | z i | = 1 , i > 2 , D ( z 1 , z 2 , . . . , z n ) � = 0 , | z 1 | ≤ 1 , | z i | = 1 , i > 1 . • [DeCarlo et al, 1977]: D ( z 1 , 1 , . . . , 1 ) � = 0 , | z 1 | ≤ 1 , D ( 1 , z 2 , 1 , . . . , 1 ) � = 0 , | z 2 | ≤ 1 , . . . . . . D ( 1 , . . . , 1 , z n ) � = 0 , | z n | ≤ 1 , D ( z 1 , . . . , z n ) � = 0 , | z 1 | = . . . = | z n | = 1 . 9/23 Yacine Bouzidi, Alban Quadrat, Fabrice Rouillier Stability of n -dimensional systems
Previous works The case n>1 • Second step: Implementation. • The case n = 2 : Numerous tests (Bistritz (94,99,02,03,04), Xu et al. 04, Fu et al. 06, etc). Most of them are based on Strintzis conditions. Generalization of univariate tests (sub-resultant computation). • The case n > 2 : very few (Dumetriscu 06, Serban and Najim 07). Sum of square techniques: either inefficient or conservative. 9/23 Yacine Bouzidi, Alban Quadrat, Fabrice Rouillier Stability of n -dimensional systems
Overview Multidimensional systems 1 Structural stability 2 Contribution on the stability test 3 Ongoing work on the stability analysis 4 10/23 Yacine Bouzidi, Alban Quadrat, Fabrice Rouillier Stability of n -dimensional systems
DeCarlo’s conditions • Start with DeCarlo’s conditions: D ( z 1 , 1 , . . . , 1 ) � = 0 , | z 1 | ≤ 1 , D ( 1 , z 2 , 1 , . . . , 1 ) � = 0 , | z 2 | ≤ 1 , . . . . . . D ( 1 , . . . , 1 , z n ) � = 0 , | z n | ≤ 1 , D ( z 1 , . . . , z n ) � = 0 , | z 1 | = . . . = | z n | = 1 . • All the conditions except the last one can be tested using classical univariate stability tests. • Focus on the condition: D ( z 1 , . . . , z n ) � = 0, | z 1 | = . . . = | z n | = 1. 11/23 Yacine Bouzidi, Alban Quadrat, Fabrice Rouillier Stability of n -dimensional systems
One first approach • z k := x k + i y k , x k , y k ∈ R , k = 1 , . . . , n , i 2 = − 1. The problem is equivalent to the study of the algebraic system: R ( D ( x 1 + i y 1 , . . . , x n + i y n )) := R ( x 1 , y 1 , . . . , x n , y n ) = 0 , C ( D ( x 1 + i y 1 , . . . , x n + i y n )) := C ( x 1 , y 1 , . . . , x n , y n ) = 0 , x 2 1 + y 2 1 − 1 = 0 , ( S ) . . . x 2 n + y 2 n − 1 = 0 . • Case n = 2 : zero-dimensional system � univariate rational representation, triangular representation, Gröbner bases. • Case n > 2 : systems with positive dimension � cylindrical algebraic decomposition, critical points methods 12/23 Yacine Bouzidi, Alban Quadrat, Fabrice Rouillier Stability of n -dimensional systems
One first approach • z k := x k + i y k , x k , y k ∈ R , k = 1 , . . . , n , i 2 = − 1. The problem is equivalent to the study of the algebraic system: R ( D ( x 1 + i y 1 , . . . , x n + i y n )) := R ( x 1 , y 1 , . . . , x n , y n ) = 0 , C ( D ( x 1 + i y 1 , . . . , x n + i y n )) := C ( x 1 , y 1 , . . . , x n , y n ) = 0 , x 2 1 + y 2 1 − 1 = 0 , ( S ) . . . x 2 n + y 2 n − 1 = 0 . • Case n = 2 : zero-dimensional system � univariate rational representation, triangular representation, Gröbner bases. • Case n > 2 : systems with positive dimension � cylindrical algebraic decomposition, critical points methods Drawback: The number of variables is doubled! 12/23 Yacine Bouzidi, Alban Quadrat, Fabrice Rouillier Stability of n -dimensional systems
Alternative approach • Goal: Avoid doubling the number of variables. • The unit poly-circle defines a n -D subspace in the 2 n -D complex space. • Via some transformations, the problem can be reduced to the search of zeros in the real space R n . • The obtained conditions are checked using classical algorithms for solving algebraic systems. 13/23 Yacine Bouzidi, Alban Quadrat, Fabrice Rouillier Stability of n -dimensional systems
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