Applied Mathematics
Checking a practical discrete-time asymptotic
stability of an interval matrix
Ayse Bulgak
Research Center of Applied Mathematics, Selcuk University, Konya, Turkey e-mail:abulgak@selcuk.edu.tr
Received: April 21, 2001
Summary.
In this paper we investigate an algorithm for checking a practical discrete-time asymptotic stability of an interval matrix.Key words:
an interval matrix, discrete-time asymptotic stability, quality of asymptotic stabilityMathematics Subject Classication (1991): 65F30, 65G10, 15A18
1. Introduction
A matrix A is said to be discrete-time asymptotically stable if the Lyapunov matrix equation A HA;A+I = 0 with the identical matrix I has a unique positive de nite solution H = H H > 0. Moreover, if A is discrete-time asymptotically stable then the value
!(A) = jjHjj
1 is known as the quality of stability for A 1],2]. For non-discrete-time asymptoticallystableAwe assume that!(A) =1. The parameter !(A) is a continuous function with respect to the entries of A. If !(A) < ! then A is known as a practical discrete-time asymptotically stable (! -stable) matrix. In 2] it was shown that the following theorem is true.
Theorem 1.
LetA be an discrete-time asymptotically stable matrix. Then for an arbitrary matrix B such thatjjBjj 1 20!3=2(A) 1 Here and everywhere in the sequel we use the notation
jjjjto designate the
the sum A+B is also an discrete-time asymptotically stable matrix and j!(A+B);!(A)j5! 5=2(A) jjBjj 0:75!(A)!(A+B)1:25!(A):
The algorithm which we suggest here is a generalization of the algorithm 3] for discrete-time asymptotic stable interval matrices.
2. Quality of discrete-time asymptotic stability of an
interval matrix
Let
A
=fA
ijg be a N-dimensional square interval matrix and the entries
A
ij = aija
ij] ofAare the intervals of the real line. The real number
!(
A
) = sup A2Af!(A)g
is known as the quality of stability of the interval matrix
A
. It is clear that for an orthogonal matrixQ we have!(
A
) =!(QA
Q):3. Practical discrete-time asymptotically stable and
practical non-discrete-time asymptotically stable interval
matrices
Let
A
=fA
ijgbe aN-dimensional square interval matrix and let! be a real number! >1. If!(
A
)< ! thenA
is said to be practical discrete-time asymptotically stable (! -stable). Otherwise,A
is said to be practical non-discrete-time asymptoticallystable (! -unstable).4. Algorithm for a matrix with only one interval entry
Let! be a real number! >1 andA
= 8 > > > < > > > : a11b11] a12 ::: a1N a21 a22 ::: a2N ... ... ... ... aN1 aN2::: aNN 9 > > > = > > >be an interval matrix with only one interval entry in position (1,1). Our aim is to nd the nite sequence of matrices. Analysis of matrices
from the sequence allows us to guarantee that the initial interval ma-trix
A
is practical discrete-time asymptotically stable or not practical discrete-time asymptotically stable.Let A= 8 > > > < > > > : a11 a12 ::: a1N a21 a22 ::: a2N ... ... ... ... aN1 aN2 ::: aNN 9 > > > = > > > : Then
A
= A+ 0b11 ;a 11] E11. Here Epq is a real matrix the element of Epq in position (pq) equals 1 and all other elements of
Epq are equal to 0.
The algorithm is as follows
1.
We assume thatA0=A.1.1.
We compute the maximal singular value N(A0) ofA0.1.2.
IfN(A0)> !1=2 then the process stops with the answer\the
given interval matrix
A
is not practical discrete-time asymp-totically stable",!(A
)> ! .1.3.
We compute h1= 1 20!3=2(A 0) and check the inequalitya11+h1b
11:If the inequality holds then the process stops with the answer
!(
A
)<1:25!(A0): Otherwise, we go to Step 2.2.
We assume thatA1=A0+h1 E11.
2.1.
We compute the maximal singular value N(A1) ofA1.2.2.
IfN(A1)> !1=2 then the process stops with the answer\the
given interval matrix
A
is not practical discrete-time asymp-totically stable",!(A
)> ! .2.3.
We compute h2= 1 20!3=2(A 1) and check the inequalitya11+h1+h2b
11:If the inequality holds then the process stops with the answer
!(
A
)<1:25 max j=01!(Aj): Otherwise, we go to Step 3.
After (k;1) steps of the process a matrixA
k ;2 and a real number
hk ;1 are computed.
k.
We assume that Ak ;1=Ak ;2+hk ;1 E11.
k.1.
We compute the maximal singular value N(Ak ;1) ofAk ;1.k.2.
If N(Ak ;1) > !1=2 then the process stops with the an-swer\the given interval matrix
A
is not practical discrete-time asymptotically stable",!(A
)> ! .k.3.
We compute hk = 1 20!3=2(A k ;1)and check the inequalitya11+h1+h2+:::+hk b
11:If the inequality holds then the process stops with the answer
!(
A
)<1:25 max j=01:::k ;1 !(Aj): Otherwise, we go to Stepk+ 1. Since hj 1=(20!3=2) it follows that the algorithm will stop after not more thenM11 steps, where
M11 = h 20! 3=2(b 11 ;a 11) i M11 is an integer:
Remark 1.The initial data of the algorithm are an integerN, a real number ! ,! >1, and the interval matrix
A
=A+ 0b11 ;a11] E
11: The algorithm has n stepsnM
11. We can compute the following real numbers 1= (
A
) = max j=01:::n;1 !(Aj) and r1=r(A
) = min j=12:::n hj r1 1 20! 3=2:The real numberr(
A
) is the smallest radius. As a result of the algo-rithm we have either the inequality!(A
)1:251or the inequality
!(
A
) > ! with the answer \the interval matrixA
is not practical discrete-time asymptotically stable".5. Algorithm for a matrix with
m
interval entries
Let! be a real number! >1 and letAbe theNN matrix with the real entries aij. We de ne an interval matrix
A
with m interval entries by the equalityA
=A+ 0b11 ;a 11] E 11+:::+ 0bij ;a ij] E ij: The matrixA
hasm interval entries in position (1,1),(2,1),:::(i; 1j), and (ij) m = i+ (j ;1)N. We include real numbers b11,
b21,:::bi;1j, and bij in the initial data of the algorithm. In the sequel we also use the algorithm for checking practical discrete-time asymptoticstability of an interval matrixwith (m;1) interval entries. For m= 1 we have considered the algorithm of such kind in section 4. The algorithm for m2 is as follows.
1.
We assume thatA
0=A+ 0b11 ;a 11] E 11+:::+ 0bi;1j ;a i;1j] E i;1j and apply the algorithm to the interval matrixA
0 with m;1 interval entries. By this way, we can compute the real numbers
1= (
A
0) and r1 =r(A
0). By the de nition, we have1 ! and r 1 1 20! 3=2: If the inequalityaij+r1 b
ij holds then we stop the process with the answer!(
A
)<1:25 1. Otherwise, we go to Step 2.2.
We assume thatA
1 =A
0+r1Eij and apply the algorithm to the interval matrixA
1 with (m;1) interval entries. By this way, we can compute the real numbers 2 = (
A
1) and r2 = r(A
1). By the de nition, we have2 ! and r 2 1 20! 3=2: If the inequalityaij+r1+r2 b
ij holds then we stop the process with the answer !(
A
) < 1:25maxf1 2
g. Otherwise, we go to Step 3.
After (p;1) steps of the process the real numbers
1 2::: p;1 and r1r2:::rp;1 are computed and for j = 12:::p
;1 we have j ! and r j 1 20! 3=2:
p.
We assume thatA
p;1=A
0+(r1+r2+:::+rp;1)Eij and apply the algorithm to the interval matrixA
p;1with (m;1) interval entries. By this way, we can compute the real numbers p = (
A
p;1) andrp =r(
A
p;1). By the de nition, we havep ! and r p 1 20! 3=2: If the inequalityaij+r1+r2+:::+rp b
ij holds then we stop the process with the answer!(
A
)<1:25maxf1 2::: p g. Other-wise, we go to Stepp+ 1. Sincerj 1=(20! 3=2) for
8jit followsthat we will stop the algorithm after not more then Mij steps, where
Mij = h 20! 3=2(b ij ;a ij) i Mij is an integer:
Remark 2.The initial data of the algorithm are an integerN, a real number ! ,! >1, and the interval matrix
A
=A+ 0b11 ;a 11] E 11+:::+ 0bij ;a ij] E ij: The algorithm hasnstepsnQ
Mij. We can compute the following real numbers = max j=12:::n j and r= min j=12:::n rj r 1 20! 3=2:
The real numberris the smallest radius. As a result of the algorithm we have either the inequality!(
A
)1:25 or the inequality!(A
)>! with the answer\the interval matrix
A
is not practical discrete-time asymptotically stable".6. Conclusion
We have suggested the computer algorithm for checking the practical discrete-time asymptotic stability of an interval matrix.
References
1. Godunov, S.K. (1998): ModernAspectsofLinear Algebra, Transl. of Math.
Monographs, 175,AMS, Providence.
2. Bulgak, H. (1999): Pseudoeigenvalues, Spectral Portrait of a matrix and their connections with dierent criteria of stability, in: Error Control and Adap-tivity in Scientic Computing, Bulgak, H. and Zenger, C. (Eds.), Kluwer Academic Publishers, 95{124.
3. Bulgak, A. (2001): Checking a practical asymptotic stability of an interval matrix. Selcuk Journal of Applied Mathematics, Vol. 2, No. 1, 17{26.
