By Mason G.
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Additional resources for 2-Dimensional vector-valued modular forms
20 G. Conte and A. M. Perdon Example 2. Let Σ be a system deﬁned over the ring R[∆] by equations of the form (3), with 00 A= , C= ∆1 . 10 The submodule S = span ∆ 0 is an (A, C) invariant submodule for Σ, since Ker C ∩ S = 0. For every R-linear map G = g1 g2 : Y → X, we have g1 ∆ g1 ∆ 2 g1 . Then (A + GC)S = span is g2 ∆ + 1 g2 (g2 ∆ + 1)∆ contained in S if and only if g2 ∆ + 1 = 0. Since this equation has no solution in R[∆], S is not an (A + GC) invariant submodule. A + GC = Conditions under which a conditioned invariant submodule is feedback invariant are given in  and in .
Massoumnia MA (1986) A geometric approach to the synthesis of failure detection ﬁlters, IEEE Trans. Automatic Control, 34:316–321 16. Proc. of the 1st IFAC Workshop on Linear Time Delay Systems (1998) Grenoble, France 17. Proc. of the 2nd IFAC Workshop on Linear Time Delay Systems (2000) Ancona, Italy 18. Proc. of the 3rd IFAC Workshop on Linear Time Delay Systems (2001) Albuquerque, NM 19. Proc. of the 4th IFAC Workshop on Linear Time Delay Systems (2003) Rocquencourt, France 20. Patton RJ, Frank PM, Clark RN (2000) Issues of fault diagnosis for dynamical systems, Springer-Verlag, New York 21.
As a result of Proposition 6, we ﬁnd that there is a part of the error dynamics, corresponding to A11 , that can possibly, under additional conditions, be modiﬁed by a suitable choice of the component g11 in G (note that the achievement of the condition (A + GC)S ∗ ⊆ S ∗ puts constraints only on the components g12 and g22 , letting the remaining ones, and in particular g11 , free). UIO and RG problems 27 Now, the problem of satisfying the condition det(sI −(A11 +g11 C11 )) ∈ H, as required to solve the UIO problem, can be reduced to that of assigning the coeﬃcients of det(sI − (A11 + g11 C11 )) and we can therefore invoke the results concerning the latter.
2-Dimensional vector-valued modular forms by Mason G.
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