奇异系统的干扰解耦(英文).pdf
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1、Journal ofM athematical Research (ii)EuAu+ I mB,uKerC. W e define the classes of subspaces T 1 ( A,E,B )= vXvKerC,AvEv+ I mB. T 2 ( A,E,B )= uXuKerC,EuAu+ I mB. T 1(A , E,B)andT2(A , E,B)are both closed under add ition.S o they have their largest m em bers.W e sym bolize the sup rem al of T 1(A , E,
2、B ), T 2(A , E,B)asT 3 1 , T 3 2seperate2 ly.T he computation of T 3 1,T 3 2is given in the next section. ForT 3 1,T 3 2w e define the classes of friends as follow s F (T 3 1 ) = F:XU(A-B F) T 3 1ET 3 1, F (T 3 2 ) = F:XUET 3 2(A-B F)T 3 2. From Theorem 2. 1 of 1, w e can easily obtain the follow in
3、g result. 425 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved. Theorem 1. 1 If (a) T 3 1KerE= 0, (b) di m(ET 3 2I mB)di m uT 3 2:A uI mB,then there ex ists a linear m ap F F (T 3 1)(FT 3 2 ), such that A2B F 2 E has linearly independent colum n f or som e comp lex num ber. Pro
4、of From Theorem 2. 1 of 1, forT 3 1,T 3 2satisfying(a) (b), there exists a linear map Fand a subspacew, w ithT 3 1wT 3 1+T 3 2such that (A-B F)wEw,ET 3 2(A-B F)T 3 2 and the matrixA2B F 2 Ehas linearly independent columns for some complex number. W e see thatwT 3 1+T 3 2KerC, AwEw+ I mB, soT 3 1A,E,
5、B ). SinceT 3 1is the supremalmember ofT1 ( A,E,B), w e havew= T 3 1. II . Conditions for the solvability of the problem: geometric characterization In this section, w e w ill study the computation ofT 3 1,T 3 2by means of subspace re2 cursins .Finally, geometric characterization for the solvability
6、 of the disturbance rejection problem for singular system sw ill be presented. W e resume our discussion by introducing a number of subspace recursions . First,W e de2 fineT (k) byT (k + 1) =A(E - 1(T(k) +I mB)KerC ), T (0) = 0.It is trivial to show that li m k T (k) exists since T (k) is monotone n
7、ondecreasing. L et this li m it beT 3. Define a second subspace sequence N (k) by N (k + 1) =T 3 +E(A - 1(N(k) +I mB)kerC ), N (0) =T 3 +I mE. N (k) is monotone nonircreasing and its li m it is denoted by N 3. Finally, define two i mpor2 tant sequences P (k) and S(k) by P (k + 1) = KerCA - 1(EP(k) +
8、I mB ), P (0) =R n, S (k + 1) = KerCE - 1(A S(k) +I mB ), S (0) =R n. W ithin at most n steps, recursion P (k) converge to T 3 1, recursion S (k) converge to T 3 2, exactly. W e have that Proposition 2. 1 N 3 =ET 3 1+AT 3 2. Proof It follow s i mmediately from the definitions thatT (k)= A S (k) for
9、allk0, Thus, w e haveT 3 =AT 3 2, and thereforeN (0) =AT 3 2+EP (0). Now assume that N (k) =AT 3 2+ EP (k). Then 525 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved. N (k + 1) =AT 3 2+E(A - 1(N(k) +I mB)KerC) =AT 3 2+EA - 1(A T 3 2+EP (k) +I mB)KerC =AT 3 2+E ( T 3 2+A - 1(EP(
10、k) +I mB)KerC =AT 3 2+ET 3 2KerC+A - 1(EP(k) +I mB)KerC (T 3 2KerC) =AT 3 2+ET 3 2+P (k + 1) =AT 3 2+ET 3 2+EP (k + 1) =AT 3 2+EP (k + 1) (ET 3 2AT 3 2 ). This proves thatN (k + 1) =A T 3 2+EP (k + 1) for all k0. Consequently, w e conclude that N 3 =AT 3 2+ET 3 1. W e can now present our result as f
11、ollow s . Theorem 2. 1 If (a) T 3 2KerE= 0, (b) di m(ET 3 2I mB)di m uT 3 2:A uI mB, (c) Ex(0- )N 3 , then the d isturbance rejection p roblem f or singular system s is solvable via state f eedback if and only ifI mSN 3 + I mB. Proof N ecessity.Taking the L aplacian transform of (1. 3a) (1. 3b )w it
12、h initial coondition Ex(0- ),w e get sE-(A-B F ) x(s ) = Ex(0 - ) +S d(s ), (2. 1a) y(s ) = Cx(s ), (2. 1b) x(s ), y(s ), d(s)are identified by the infinite sequences x-,x-+ 1,. . . ,x0,x1,. . . , y-,y-+ 1,. . . ,y0,y1,. . . , d- 1,d-,d-u- 1,. . . ,d0,d1,. . . separately, defined by 9. x(s ) = x-s+x
13、-+ 1s- 1+ . . . +x- 1s+x0+x1s- 1+x2s- 2+ . . . , y(s ) = y-s+y-+ 1s- 1+ . . . +y- 1s+y0+y1s- 1+y2s- 2+ . . . , d(s ) = d- 1s+ 1+d-s+d-+ 1s- 1+ . . .d- 1s+d0+d1s- 1+d2s- 2+ . . . , (2. 2) (2. 3) (2. 4) w hen no confusion is possible,w e abuse the term inology and refer tox(s ), y(s ), d(s)or to their
14、 laurent expansions as the trajectory, the output and the disturbance generated by the initial conditionEx(0- ). Now , disturbance rejection demands for someF, for anyEx(0- )N 3 , the L aurent expansions ofx(s ), y(s ), d(s)w hich satisfy (2. 1a) (2. 1b )satisfy the follow ing equations for some: 62
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