Controllability Analysis of Linear Discrete Time Systems with Time Delay in State

and Applied Analysis 3 Definition 2.1 symmetrical subspace . Given matrices A,D ∈ Rn×n, one defines a subset of R n×n as follows: P A,D : { f A,D | f A,D M ∑ m 1 cmA m,1Dm,1 · · ·Aim,nmDjm,nm , nm ∈ N, im,1, jm,1, . . . , im,nm , jm,nm ∈ N, 0 ≤ im,1, jm,1, . . . , im,nm , jm,nm ≤ n, cm ∈ R, m 1, . . . ,M; M ∈ N } . 2.1 Then a one-to-one mapping Θ : P A,D → P A,D is constructed as follows: give any f A,D ∈ P A,D , supposing f A,D ∑Mm 1 cmAm,1Dm,1 , . . . , Am,nmDm,nm , one defines Θ ( f A,D ) : M ∑ m 1 cm ( Am,1Dm,1 · · ·Aim,nmDjm,nm Dm,nmAm,nm · · ·Djm,1Aim,1 ) . 2.2 Then we get a subspace of P A,D by PSym A,D : { Θ ( f A,D ) | ∀f A,D ∈ P A,D . 2.3 We call PSym A,D the symmetrical subspace of P A,D . By Definition 2.1, it is easy to verify that I,A,D,AD DA,ADA,DAD ∈ PSym A,D , where I is the identity matrix in Rn×n. Now, we introduce a matrix sequence {Gk}k 0 ∈ Rn×n as follows: Gk { A, if k 0, 1, . . . , h, Gk−1A Gk−1−hD, if k h 1, h 2, . . . 2.4 Then the solution of system 1.1 can be expressed as x k 1 Ψk x 0 , . . . , x −h k ∑ i 0 Gk−iBu i , k > 0, 2.5 where Ψk x 0 , . . . , x −h is the part of the solution with zero input. Lemma 2.2. Given the matrix sequence {Gk}k 0 in 2.3 , the following statements hold: a there are no similar terms between GkA and Gk−hD; b for any Gk, it can be expressed as Gk ∑ ∀f A,D ∈Qk A,D f A,D , 2.6 4 Abstract and Applied Analysis where Qk A,D is a subset of Rn×n defined as follows: Qk A,D { f A,D | f A,D A1D1 · · ·AikDjk , ∀i1, j1, . . . , ik, jk ∈ N, k ∑ m 1 im h 1 k ∑ m 1 jm k } ; 2.7 c for all k ∈ N, Gk ∈ PSym A,D . Proof. Statement a is nearly self-evident because any term fromGkA is ended byA, whereas any term from Gk−hD is ended by D. To prove the result of statement b , mathematical induction is invoked. I The first h 1 terms of {Gk}k 0 are I,A, . . . , A, and it is obvious that Qk A,D {Ak}, for k 0, 1, . . . , h. Thus, for k 0, 1, . . . , h, 2.5 holds. II Assume that, for l k, k 1, . . . , k h, Gl is expressed in form 2.5 . We will prove that Gk h 1 can be expressed in form 2.5 as well. First, we prove that, for any k, we have Qk 1 A,D Qk A,D A ⋃ Qk−h A,D D, 2.8 where the sets Qk A,D A, Qk−h A,D D are defined as Qk A,D A { f A,D A | f A,D ∈ Qk A,D } , Qk−h A,D D { f A,D D | f A,D ∈ Qk−h A,D } . 2.9 By the assumption, we have Qk A,D A { f A,D A | f A,D ∈ Qk A,D } { f A,D A | f A,D A1D1 · · ·AikDjk , i1, j1, . . . , ik, jk ∈ N, k ∑ m 1 im h 1 k ∑ m 1 jm k } , Abstract and Applied Analysis 5 { g A,D | g A,D A1D1 · · ·Aik 1Dk 1 , i1, j1, . . . , ik, jk ∈ N,and Applied Analysis 5 { g A,D | g A,D A1D1 · · ·Aik 1Dk 1 , i1, j1, . . . , ik, jk ∈ N, ik 1 1, jk 1 0, k 1 ∑ m 1 im h 1 k 1 ∑ m 1 jm k 1 } ⊆ Qk 1 A,D , Qk−h A,D D, { f A,D D | f A,D ∈ Qk−h A,D } { f A,D D | f A,D A1D1 · · ·Aik−hDjk−h , i1, j1, . . . , ik−h, jk−h ∈ N, k−h ∑ m 1 im h 1 k−h ∑ m 1 jm k − h } { g A,D | g A,D A1D1 · · ·Aik 1Dk 1 , i1, j1, . . . , ik−h, jk−h ∈ N, ik−h 1 · · · ik 1 0, jk−h 1 · · · jk 0, jk 1 1, k 1 ∑ m 1 im h 1 k 1 ∑ m 1 jm k 1 }