J. Sreedhar and P. Van Dooren, Periodic descriptor systems: Solvability and conditionability, IEEE Trans. Auto. Control, 44(1999), 310-313.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... state space dimension n and or large period appear, e.g. in the helicopter ground resonance damping problem and the satellite attitude control problem; see, e.g. 9, 22, 26, 37] The analysis and design of these class of systems has received considerable attention in recent years (see, e.g. [10, 11, 13, 26, 33, 32, 36, 37]) The need for parallel computing in this area can be seen from the fact that (2) represents a non linear system with pn 2 unknowns. Reliable methods for solving these equations have a computational cost in flops (floating point arithmetic operations) of O(pn 3 ) In this paper we analyze ....

J. Sreedhar and P. Van Dooren. Periodic descriptor systems: Solvability and conditionability. IEEE Trans. Automat. Control, 44(2):310--313, 1999.

.... It has already been observed in the introduction that the solution of the difference equation E k x k 1 = A k x k , x 0 = x 0 is related via monodromy matrices of the form (1) If the coefficient matrices are p periodic as in (4) then the difference equation is solved (if solvable, see [20, 26]) by x j p = p Gamma1 Y k=0 (E k j nA k j ) x j ; j = 0; 1; for given starting values x 0 , x 1 , x p Gamma1 . Here the notation y = EnA)x is short for (y; x) 2 (EnA) Due too the periodicity of the coefficients, there are p different products needed to define the whole ....

J. Sreedhar and P. Van Dooren. Periodic descriptor systems: Solvability and conditionability. IEEE Trans. Automat. Control, 44(2):310--313, 1999.

....B k K , for all k 2 N , and K is the smallest positive integer for which this holds. If we allow both the E k and A k matrices to be singular, then x k may still uniquely be defined in the context of a two point boundary value problem, as e.g. in optimal control of periodic systems. It is shown in [4] that a necessary and sufficient condition for this is that the pencil E Gamma A : 2 6 6 6 4 E 1 GammaA 1 GammaA 2 E 2 . GammaA K EK 3 7 7 7 5 (1.2) is regular (i.e. det(E Gamma A) j= 0) We call such periodic systems regular. We can always define a periodic similarity ....

J. Sreedhar, P. Van Dooren, "Periodic descriptor systems : solvability and conditionability ", to appear in the IEEE Trans. Aut. Contr., 1997. 4

....with a boundary condition B 0 x 0 = BN xN b: 4) Before answering that question, we first introduce some notation. Let us call S(0; N Gamma 1) the matrix in the left hand side of (2) and C(0; N Gamma 1) its submatrix obtained after deleting its first and last block columns. It was shown in [6] that ffl if S(0; N Gamma 1) has full row rank, then there is an n dimensional subspace of solutions x(0; N) to the equation (2) for every input sequence u(0; N Gamma 1) since the image of S(0; N Gamma 1) is the whole space and its kernel has dimension n) we then say that (2) is solvable ....

....= n. Sufficient conditions will be given later. 2 Periodic boundary value problems A periodic system is a set of difference equations : E k x k 1 = A k x k u k ; k = 0; 6) with A k = AK k ; E k = EK k ; 8k, and the period is the smallest value of K for which this holds. It was shown in [6] that a periodic system of period K is solvable and conditionable for all N provided the pencil E Gamma A : 2 6 6 6 6 4 GammaA 0 E 0 . GammaA K Gamma2 EK Gamma2 EK Gamma1 GammaA K Gamma1 3 7 7 7 7 5 (7) is regular (i.e. det(E Gamma A) j= 0) We then call the periodic ....

J. Sreedhar and P. Van Dooren, "Periodic descriptor systems : solvability and conditionability ", to appear in the IEEE Trans. Aut. Contr., 1998.

....initiated by the Belgian State, Prime Minister s Office for Science, Technology and Culture. The scientific responsibility rests with its authors. condition 2 6 6 6 4 E 0 E 1 . EK Gamma1 3 7 7 7 5 Gamma 2 6 6 6 4 A 0 A 1 . AK Gamma1 3 7 7 7 5 (2) is regular. Recently [1] we have shown that statement (2) is equivalent to solvability or conditionability of Sigma, an assumption commonlymade in connection with the Two point Boundary Value Problem (TPBVP) 2 6 6 6 6 6 6 6 4 GammaA(0) E(0) GammaA(1) E(1) GammaA(N Gamma 1) E(N Gamma 1) ....

J. Sreedhar and P. Van Dooren, "Periodic descriptor systems: Solvability and conditionability." presented at MTNS, St. Louis, MO, USA, June 24--28, 1996.

....3.1. PERIODIC BOUNDARY VALUE PROBLEMS A periodic system is a set of difference equations (16) where now the coefficient matrices vary periodically with time, i.e. M k = M k K ; 8k and for M = E; A; B; C and D. The period is the smallest value of K for which this holds. It was shown in [18] that a periodic system of period K is solvable and conditionable (i.e. has a well defined solution for suitably chosen boundary conditions F 1 ; FN ) for all N , provided the pencil E Gamma A : 2 6 6 6 4 GammaA 1 E 1 . GammaA K Gamma1 EK Gamma1 EK GammaA K 3 7 7 7 5 ....

J. Sreedhar and P. Van Dooren (1999), Periodic descriptor systems: Solvability and conditionability, IEEE Trans. Aut. Contr., to appear.

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J. Sreedhar and P. Van Dooren, Periodic descriptor systems: Solvability and conditionability, IEEE Trans. Auto. Control, 44(1999), 310-313.

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