Abstract
Alternative methods are proposed for test of output feedback stabilizability and construction of a stable closed-loop polynomial for 2D systems. By the proposed methods, the problems can be generally reduced to the 1D case and solved by using 1D algorithms or Gröbner basis approaches. Another feature of the methods is that their extension to certain specialnD (n>2) cases can be easily obtained.
Moreover, the “Rabinowitsch trick,” a technique ever used in showing the well-known Hilbert's Nullstellensatz, is generalized in some sense to the case of modules over polynomial ring. These results eventually lead to a new solution algorithm for the 2D polynomial matrix equationD(z, w)X(z, w)+N(z, w)Y(z, w)=V(z, w) withV(z, w) stable, which arises in the 2D feedback design problem. This algorithm shows that the equation can be effectively solved by transforming it to an equivalent Bezout equation so that the Gröbner basis approach for polynomial modules can be directly applied.
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Abbreviations
- R :
-
the field of real numbers
- C :
-
the field of complex numbers
- R[z, w]:
-
commutative ring of 2D polynomials inz andw with coefficients inR
- M(R[z, w]):
-
set of matrices with appropriate dimensions with entries inR[z, w]
- R[z, w]n :
-
module of orderedn-tuples inR[z, w]
- R[z, w]n ×m :
-
set ofn ×m matrices with entries inR[z, w]
- Ū :
-
closed unit disc inC, i.e., {z ɛC| |z| ≤ 1}
- Ū 2 :
-
closed unit bidisc, i.e., {(z, w) ɛC 2| |z| ≤ 1, |w| ≤ 1}
- A T :
-
transpose of matrixA
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Xu, L., Saito, O. & Abe, K. Output feedback stabilizability and stabilization algorithms for 2D systems. Multidim Syst Sign Process 5, 41–60 (1994). https://doi.org/10.1007/BF00985862
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DOI: https://doi.org/10.1007/BF00985862