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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
15、5 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved. Ex-=S d- 1,Cx-= 0, Ex- 1=(A-B F)x-+S d-,Cx- 1= 0, . . . . . . Ex- 1=(A-B F)x- 2+S d- 2,Cx- 1= 0, Ex0=(A-B F)x- 1+S d- 1,Cx0= 0, Ex1=(A-B F)x0+Ex(0 - ) +S d0,Cx1= 0, Ex2=(A-B F)x1+S d1,Cx2= 0, . . . . . . Note thatP (0)= R n, f
16、rom the first equation w e get that S d- 1=Ex-=A(A - 1Ex -)AT 3 1ET 3 1+I mB. Form the second equation,w e get S d-=Ex- 1-(A-B F)X- =A(A - 1EX -+ 1 ) - EE - 1(A X -B Fx ) AT 3 1+ET 3 2ET 3 1+AT 3 2+I mB. By the same w ay,w e knowS dkET 3 1+AT 3 2+ I mB f or all k- 1, so I mSET 3 1+AT 3 2+I mB. Suffi
17、ciency.Suppose T 3 1,T 3 2satisfy(a) (b), from Theorem 1. 1, w e can findFT (T 3 1)T(T 3 2)such that(A-B F)T 3 1ET 3 1, ET 3 2(A-B F)T 3 2.andA-B F- Ehas linearly independent columns for some complex number. By theorem 1 of 3, from the matrixA-B F-Ehaving linearly independent columns for some comple
18、x number,w e can reach the uniqueness of solution to the closed2loop system (1. 3a ). thus it is sufficient to consider existence of solutions . since I mSET 3 1+AT 3 2+ I mB, according to Theorem 3. 2 of 1,w e know (1. 3a) (1. 3b )has a solutionx(t), furthermorex(t)T 3 1+T 3 2KerC w hich meansy(t )
19、= Cx(t )= 0. This completes the proof. The proof of follow ing lemma is obvious and is om itted. Lemma Given S1,B and a subspacevX,there ex ists F1:DX such that I m ( S1-B F1)I mB+ v if and only ifI mS1I mB+v. A ccord ing to the above lemm a and T heorem2. 1,w e can obtain our f inal result easily.
20、Theorem 2. 2 If (a) T 3 1KerE= 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 f or singular system s via linear state f eedback and linear f eedf or2 725 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved. w ard control ofthe d isturb
21、ances is solvable if and only ifI mS1N 3 + I mB. III . Conclusion W e have provided necessary and sufficent condition for the solvability of disturbance re2 jection for singular system s . W e have taken care to ensure that the resulting closed2loop sys2 tem has smooth solutions for a w ide class of
22、 disturbance functions and initial conditions and that. W hen a solution exists, it is unique. Ourmode of argument has been in the spirit ofW onham 2 and the structure of our proof clearly resembles that of the corresponding result in the state space case. W e have found the usual presentation, in t
23、erm s of the properties of uniquely edterm ined maxi mal elements of certain collections of subspaces, by means of some subspace recursions . This means that our result is constructive and algorithm ic. Throughout this paperw e have had in m ind a linear system described by a m ixture of al2 gebraic
24、 and differential equations, the extension of these results to the discrete2ti me case should not present any major difficulty. References 1 FletcherL R andA asaraaA.On d isturbance decoup ling in descrip tor system s. SI AM J. Control andOpti2 m ization, 1989, 27(6): 1319- 1332 2 Wonham W M.L inear
25、 M ultivariable Control;A Geom etric A pp roach.N ew York: Springer- V erlag, 1979. 3 Fleltcher L R.R egularisability of descrip tor system s.Internat. J. System s Sci . , 1986, 17: 834- 847 4 Zheng Z, Shayman M A and Tarn T J.S ingular system s:a newapp roach in the tim e dom ain.IEEE T rans . A ut
26、omat. Control, 1987, AC-32(1): 42- 50 5 U lviye Baser and Kadri Ozcaldiran.On Observability ofsingular system s.Circuits System s Signal Pro2 cess, 1992, 11(3): 421- 430 6 Beauchamp G, Banaszuk A , KocieckiM and L ew is F L.Inner and outer geom etry f or singular system s w ith computation of subspc
27、es.I N T. J. CON TROL , 1991, 53(3): 661- 687 7 Wong K T.T he eigenvalue p roblemT x+S x. J. D ifferential Equations, 1974, 16: 270- 280 8 L ew is F L.A survey oflinear singular system s. J. Circuits System s Signal Process, 1986, 5(1): 3- 36 9 M alabreM.M ore geom etry about singular system s.Proc.
28、 26th IEEE CDC. Los A ngeles, 1987, 1138- 1139 10 Bhattacharyya S P.D isturbance rejection in linear system s.I N T. J. SYSTEM S SC I . , 1974, 5(7): 633- 637 奇 异 系 统 的 干 扰 解 耦 范文涛 谭连生 (中国科学院武汉数学物理研究所,武汉430071) 摘 要 本文针对线性的不变奇异系统,讨论了利用一般状态反馈及前馈控制的组合来达到其干 扰解耦的条件.文中构造性地给出了这一问题可解的充分必要条件. 825 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.