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Appendix A Elementary Operations on Matrices, Matrix Equations with Nonnegative Solution and Shuffle Algorithm Ele mentary Operations on Matrices, Matrix Equations w ith Nonnegative Solution
A.1 Elementary Operations on Numerical Matrices
Definition A.1. The following operations are called elementary operations on real
(complex) matrices mnA ×ℜ∈ ( mnA ×∈C ):
1) Multiplication of any i-th row (column) by a real number c ≠ 0. This operation will be denoted by L[i×c] (R[i×c]).
2) Addition to any i-th row (column) of A its j-th row (column) multiplied by a real number c ≠ 0. This operation will be denoted by L[i+j×c] (R[i+j×c]).
3) Interchange of any two rows (columns) of A e.g. i-th and j-th rows (columns). This operation will be denoted by L[i,j] (R[i,j]).
It is easy to verify that the above elementary operations performer on rows are equivalent to pre-multiplication of the matrix A by the following matrices:
rowth
rowth
,
0...
...
0...
...
0...
0...
0...0...00
......
...1...00
...
0...0...10
0...0...01
),,(
,
1...0...00
...
0......00
...
0...0...10
0...0...01
),(
columnth
−
−
×
×
−
ℜ∈
=
ℜ∈
=
j
i
nn
ji
a
nn
i
m
ccjiL
cciL
480 A Elementary Operations on Matrices, Matrix Equations
.
1...0...0...00
.........0...
...
0...
...
0...
0...
0...1...00
......
1...0...00
......
0...0...10
0...0...01
),( nn
ji
i jiL ×ℜ∈
=
(A.1)
The same operations carried out on columns are equivalent to post-multiplication of the matrix A by the following matrices:
rowth
rowth
,
0...0...0...00...
0...
0...
...
0...
0...
......
1......00
0...1...00
...
0...0...10
0...0...01
),,(
,
1...0...00
...
0......00
...
0...0...10
0...0...01
),(
columnth
−
−
×
×
−
ℜ∈
=
ℜ∈
=
j
i
mm
ji
a
mm
i
m
c
cjiR
cciR
A.2 Elementary Operations on Polynomial Matrices 481
.
1...0...0...00
.........0...
...
0...
...
0...
0...
0...1...00
......
1...0...00
......
0...0...10
0...0...01
),( mm
ji
i jiR ×ℜ∈
=
(A.2)
It is easy to verify that the determinants of the matrices and (A.2) are nonzero.
Lemma A.1. The elementary operations performed on the matrix A does not change its rank.
Proof. It is easy to verify that the elementary operations do not change minors of the matrix A. □
A.2 Elementary Operations on Polynomial Matrices
Definition A.2. The following operations are called elementary operations
on a polynomial matrix ][)( ssA mn×∈C :
1) Multiplication of any i-th row (column) of A(s) by the number c ≠ 0. This operation will be denoted by L[i×c] (R[i×c]).
2) Addition to any i-th row (column) of A(s) its j-th row (column) multiplied by a polynomial w(s) ≠ 0. This operation will be denoted by L[i+j×w(s)] (R[i+j×w(s)]).
3) Interchange of any two rows (columns) of A(s) e.g. i-th and j-th rows (columns). This operation will be denoted by L[i,j] (R[i,j]).
It is easy to verify that the above elementary operations performer on rows are equivalent to pre-multiplication of the matrix A(s) by the following matrices:
482 A Elementary Operations on Matrices, Matrix Equations
.
1...0...0...00
.........0...
...
0...
...
0...
0...
0...1...00
......
1...0...00
......
0...0...10
0...0...01
),(
],[
0...
...
0...
...
0...
0...
0...0...00
......
)(...1...00
...
0...0...10
0...0...01
))(,,(
,
1...0...00
...
0......00
...
0...0...10
0...0...01
),(
rowth
rowth
columnth
nn
ji
i
nn
ji
a
nn
i
m
jiL
ssw
swjiL
cciL
j
i
×
×
×
−
ℜ∈
=
ℜ∈
=
ℜ∈
=
−
−
(A.3)
The same operations performed on columns are equivalent to post-multiplication of the matrix A(s) by the following matrices:
A.2 Elementary Operations on Polynomial Matrices 483
.
1...0...0...00
.........0...
...
0...
...
0...
0...
0...1...00
......
1...0...00
......
0...0...10
0...0...01
),(
],[
0...0...0...00...
0...
0...
...
0...
0...
......
1...)(...00
0...1...00
...
0...0...10
0...0...01
))(,,(
,
1...0...00
...
0......00
...
0...0...10
0...0...01
),(
rowth
rowth
columnth
mm
ji
i
mm
ji
a
mm
i
m
jiR
s
sw
swjiR
cciR
j
i
×
×
×
−
ℜ∈
=
ℜ∈
=
ℜ∈
=
−
−
(A.4)
It is easy to verify that the determinants of the matrices (A.3) and (A.4) are nonzero and do not depend on the variable s. Such matrices are called unimodular matrices.
Lemma A.2. The elementary operations performed on the matrix A(s) does not change its rank.
Proof is similar to the proof of Lemma A.1.
484 A Elementary Operations on Matrices, Matrix Equations
A.3 Matrix Equations with Nonnegative Solutions
Consider the matrix equation
bAx = (A.5)
where nnA ×ℜ∈ , nb ℜ∈ . It is assumed that the equation (A.5) has a solution, i.e.
AbA rank ][rank = (A.6)
Theorem A.1. Let the assumption (A.6) be satisfied. The equation (A.5) has
a nonnegative solution nx +ℜ∈ if
01
≥=
r
i i
iTT
i
s
buAu for all 0≥is , ri ,...,2,1= (rank of ATA) (A.7)
where si is an eigenvalue of ATA and ui is its eigenvector associated with si, i.e.
niusAuA iiiT ,...,2,1, == (A.8)
and 1=iu .
Proof. Premultiplying the equation (A.5) by AT we obtain
bAAxA TT = . (A.9)
Premultiplication of equation (A.9) by Tiu yields
nibAuAxAu TTi
TTi ,...,2,1, == (A.10)
and using equation (A.8) we obtain
nibAuxus TTi
Tii ,...,2,1, == . (A.11)
Taking into account that si = 0 for i = r + 1,2,…,n from equation (A.11) we obtain
==
==r
i i
iTT
ir
ii
Ti s
buAuxuux
11 for all 0>is , ri ,...,2,1= . (A.12)
since 1== iiTi uuu .
Therefore, the equation (A.5) has a nonnegative solution nx +ℜ∈ if the
condition (A.7) is satisfied. □
A.3 Matrix Equations with Nonnegative Solutions 485
Definition A.3. A matrix nnijaA ×ℜ∈= ][ is called the matrix with cyclic
structure (shortly cyclic matrix) if and only if
iiiiniiii aaaaa ,1,1,,1, ...... −+ ≥≥≥≥≥≥ for ni ,...,2,1= (A.13)
and strictly cyclic if and only if
iiiiniiii aaaaa ,1,1,,1, ...... −+ >>>>>> . (A.14)
Lemma A.3. A cyclic matrix A with nonnegative entries has nonnegative determinant, i.e. det A ≥ 0.
Proof is given in [175].
Definition A.4. The equations (A.5) and
bxA = , nnA ×ℜ∈ , nb ℜ∈ (A.15)
are called equivalent if and only if the equation (A.15) is obtained form equation (A.5) by performing the following two elementary operations:
multiplication the i-th equation (i = 1,2,…,n) of (A.5) by a nonzero number c, addition to the i-th equation the j-th equation of (A.5) multiplied by a nonzero number c.
It is well-known that the equivalent equations (A.5) and (A.15) have the same solution x.
Theorem A.2. The equation (A.5) with nnA ×+ℜ∈ and nTb +ℜ∈= ]1...1[
has nonnegative (positive) solution nx +ℜ∈ if and only if the matrix A is cyclic
(strictly cyclic). Proof is given in [175].
Remark A.1. In many cases if the matrix A of the equation (A.5) has some
negative entries we may reduce it to the desired form with nnA ×+ℜ∈ by the use
of the elementary operations defined in Definition A.1.
Example A.1. Consider the equation
=
−
1
1
1
101
001
012
3
2
1
x
x
x
. (A.16)
To eliminate the negative entry a12 = – 1 in the matrix A we subtract from the second equation the first one of (A.16) and we get
486 A Elementary Operations on Matrices, Matrix Equations
=
−
−
1
0
1
101
011
012
3
2
1
x
x
x
. (A.17)
Next we add to the first equation the second one of (A.17) and we obtain
=
−
1
0
1
101
011
001
3
2
1
x
x
x
. (A.18)
By removing from (A.18) the second equation and taking into account that x1 = x2 we obtain the equation
=
1
1
11
01
3
1
x
x (A.19)
with cyclic matrix
=
11
01A and x1 = 1, x3 = 0.
A.4 Transformation of the State Equations by the Use of t he Shuffle Algorithm
A.4 Transformation of the State Equations by the Use of the Shuffle Algorithm
A.4 Transformation of the State Equations by the Use of t he Shuffle Algorithm
A.4.1 Continuous-Time Linear Systems
Using the shuffle algorithm [101] we shall transform the state equation )()()( tButAxtxE += with 0det =E and regular pencil AEs − to the
equivalent form )(...)()()()( )1(1
)1(10 tuBtuBtuBtxAtx q
q−
−++++= .
Performing elementary row operations on the array
BAE (A.20)
or equivalently on )()()( tButAxtxE += we get
22
111
0 BA
BAE
(A.21)
and )()()( 111 tuBtxAtxE += , (A.22)
)()(0 22 tuBtxA += , (A.23)
where 1E has full row rank. Differentiation of (A.23) with respect to time yields
A.4 Transformation of the State Equations by the Use of the Shuffle Algorithm 487
)()( 22 tuBtxA −= . (A.24)
The equations (A.22) and (A.24) can be written in the form
)(0
)(0
)(0
)(2
11
2
1 tuB
tuB
txA
txA
E
−
+
+
=
. (A.25)
The array
22
111
00
0
BA
BAE
−
(A.26)
can be obtained from (A.21) by performing a shuffle. If matrix
2
1
A
E
(A.27)
is nonsingular then solving (A.25) we obtain
−
+
+
=
−)(
0)(
0)(
0)(
2
111
2
1 tuB
tuB
txA
A
Etx
.
(A.28)
If the matrix (A.27) is singular then performing elementary row operations on (A.26) (or equivalently on (A.25)) we obtain
244
1332
0 CBA
CBAE
(A.29)
and
)()()()( 1332 tuCtuBtxAtxE ++= , (A.30)
)()()(0 244 tuCtuBtxA ++= , (A.31)
where 2E has full row rank and 12 rankrank EE ≥ . Differentiation of (A.31)
with respect to time yields
)()()( 244 tuCtuBtxA −−= . (A.32)
The equations (A.30) and (A.32) can be written in the form
)(0
)()(0
)(0
)(24
133
4
2 tuC
tuB
Ctu
Btx
Atx
A
E
−
+
−
+
+
=
. (A.33)
488 A Elementary Operations on Matrices, Matrix Equations
The array
244
1332
00
0
CBA
CBAE
−−
(A.34)
can be obtained from (A.29) by performing a shuffle. If matrix
4
2
A
E (A.35)
is nonsingular, we can solve (A.33) in a similar way to (A.25). If the matrix (A.35) is singular, we repeat the procedure for (A.34). After q – 1 steps we obtain a nonsingular matrix
+
−
1
1
q
q
A
E
(A.36)
and
−
++
+
+
= −
−
+
−)(
0...)(
0)(
0)(
0)( )1(
1
1
1tu
Htu
Ctu
Btx
A
A
Etx q
q
qqq
q
q ,
(A.37)
where kkk dttudtu /)()()( = . From the above considerations, we have the
following procedure. Procedure A.1 Step 1. Performing elementary row operations on (A.20) gives (A.21) where 1E
has full row rank. Step 2. Shuffle array (A.21) to (A.26). If the matrix
2
1
A
E
(A.38)
is nonsingular, find the desired solution from (A.25). If the matrix is singular, performing elementary row operations on (A.26) gives (A.29). If the pencil is regular, by Step 1 and Step 2 we finally obtain a regular system (A.37).
Remark A.2. Using the shuffle algorithm we may also find the index q of the pair (E, A). The index is equal to the number of performed shuffles to find (A.37).
Example A.2. Using Procedure A.1 transform the equation
A.4 Transformation of the State Equations by the Use of the Shuffle Algorithm 489
)(
10
01
00
)(
010
010
001
)(
100
110
010
tutxtx
−+
=
−
(A.39)
to the form )(...)()()()( )1(1
)1(10 tuBtuBtuBtxAtx q
q−
−++++= and find
the index q of the pair (E, A).
Step 1. Performing on the array
10010000
01010110
00001010
−−=BAE
(A.40)
the elementary row operations ]112[ ×+L , ]3,2[L we obtain
01011000
10010100
00001010
0 22
111
−=
BA
BAE. (A.41)
Step 2. Performing the shuffle on (A.41) we get
0100000011
0010010100
0000001010
00
0
22
111 =− BA
BAE. (A.42)
The matrix
=
011
100
010
2
1
A
E
(A.43)
is nonsingular and from (A.25) we obtain
).(
00
00
01
)(
10
00
00
)(
010
001
001
)(0
)(0
)(0
)(2
111
2
1
tututx
tuB
tuB
txA
A
Etx
+
+
−=
−
+
+
=
−
(A.44)
In this case we have performed only one shuffle. Therefore, the index q is equal one.
490 A Elementary Operations on Matrices, Matrix Equations
A.4.2 Discrete-Time Systems
In a similar way as for continuous-time systems using the shuffle algorithm we shall transform the state equation iii BuAxEx +=+1
with 0det =E and regular pencil AEz − to the equivalent form
111101 ... −+−++ ++++= qiqiiii uBuBuBxAx .
Performing elementary row operations on the array (A.20) (or equivalently on iii BuAxEx +=+1 ) we have (A.21) and
iii uBxAxE 1111 +=+ , (A.45)
ii uBxA 220 += , (A.46)
where 1E has full row rank. Substituting in (A.46) i by i + 1 we obtain
1212 ++ −= ii uBxA . (A.47)
The equations (A.45) and (A.47) can be written in the form
1
2
111
2
1 0
00 ++
−
+
+
=
iiii u
Bu
Bx
Ax
A
E. (A.48)
The array
22
111
00
0
BA
BAE
−
(A.49)
can be obtained from (A.21) by performing a shuffle. If matrix (A.27) is nonsingular then solving (A.48) we obtain
−
+
+
= +
−
+ 12
111
2
11
0
00 iiii uB
uB
xA
A
Ex . (A.50)
If the matrix (A.27) is singular then performing elementary row operations on (A.49) we obtain (A.29) and
113312 ++ ++= iiii uCuBxAxE , (A.51)
12440 +++= iii uCuBxA , (A.52)
where 2E has full row rank and 12 rankrank EE ≥ . Substituting in (A.52)
i by i + 1 we obtain
221414 +++ −−= iii uCuBxA . (A.53)
A.4 Transformation of the State Equations by the Use of the Shuffle Algorithm 491
The equations (A.51) and (A.53) can be written as
2
21
4
1331
4
2 0
00 +++
−
+
−
+
+
=
iiiii u
Cu
B
Cu
Bx
Ax
A
E. (A.54)
The array
244
1332
00
0
CBA
CBAE
−−
(A.55)
can be obtained from (A.29) by performing a shuffle. If matrix (A.35) is nonsingular, we can find 1+ix from (A.54). If the matrix is singular, we repeat the
procedure for (A.55). If the pencil is regular then after q – 1 steps we obtain a nonsingular matrix
+
−
1
1
q
q
A
E
(A.56)
and
−
++
+
+
= −++
−
+
−+ 11
1
1
11
0...
000 qiq
iq
iq
iq
q
qi u
Hu
Cu
Bx
A
A
Ex . (A.57)
Example A.3. Using Procedure A.1 transform the equation
iii uxx
+
−−=
−−+
02
10
01
322
100
011
022
020
001
1
(A.58)
to the form 111101 ... −+−++ ++++= qiqiiii uBuBuBxAx and find the index q
of the pair (E,A).
Step 1. Performing on the array
02322022
10100020
01011001
−−−−=BAE
(A.59)
the elementary row operations ]213[ ×+L , ]123[ ×+L we obtain
492 A Elementary Operations on Matrices, Matrix Equations
14200000
10100020
01011001
0 22
111
−=
BA
BAE. (A.60)
Step 2. Performing the shuffle on (A.60) we get
1400000200
0010100020
0001011001
00
0
22
111 =− BA
BAE. (A.61)
The matrix
=
200
020
001
2
1
A
E
(A.62)
is nonsingular and from (A.50) we obtain
.
5.02
00
00
00
5.00
01
000
5.000
011
0
00
1
12
111
2
11
+
+
−
+
+
+
=
−
+
+
=
iii
iiii
uux
uB
uB
xA
A
Ex
(A.63)
In this case the index q is also equal one.
Appendix B Positive Regular Continuous-Time and Discrete-Time Linear Systems
Positive Regular Cont inuous-Time and Discrete-Time Linear Syste ms
B.1 Externally Positive Continuous-Time Linear Systems
Consider the continuous-time linear system described by the equations
),()()(
,)0(),()()( 0
tDutCxty
xxtButAxtx
+==+=
(B.1)
where ntx ℜ∈)( is the state vector, mtu ℜ∈)( is the input vector and
pty ℜ∈)( is the output vector and nnA ×ℜ∈ , mnB ×ℜ∈ , npC ×ℜ∈ ,
mpD ×ℜ∈ .
Definition B.1. The system (B.1) is called externally positive if for every input mtu +ℜ∈)( and x0 = 0 the output pty +ℜ∈)( for all t ≥ 0.
The impulse response g(t) of single-input single-output system is called the output of the system for the input equal to Dirac impulse δ(t) with zero initial conditions. In a similar way assuming successively that only one input is equal to δ(t) and the remaining are zero we may define the matrix of impulse responses
mptg ×ℜ∈)( of a system with m-inputs and p-outputs.
Theorem B.1. The system (B.1) is externally positive if and only if its matrix of impulse responses is nonnegative, i.e.
mptg ×
+ℜ∈)( for all t ≥ 0. (B.2)
Proof. The necessity of condition (B.2) follows immediately from Definition B.1. The output of the system (B.1) with zero initial conditions for any input u is given by the formula
−=t
dutgty0
)()()( τττ . (B.3)
494 B Positive Regular Continuous-Time and Discrete-Time Linear Systems
If the condition (B.2) is satisfied and mtu +ℜ∈)( then from (B.3) we have
pty +ℜ∈)( for t ≥ 0. □
Theorem B.2. The continuous-time system described by the differential equation (the ARMA model)
−
=
−
==+
1
0
)(1
0
)()(n
i
ii
n
i
ii
n ubyay
(B.4)
is externally positive if
ai ≤ 0 and bi ≥ 0 for i = 0,1,…, n – 1, (B.5)
where ,:)(i
ii
dt
ydy = ,:)(
i
ii
dt
udu = i = 0,1,…, n-1.
Proof. We shall show that if the conditions (B.5) are satisfied then +ℜ∈)(tg
for all t ≥ 0. Applying the Laplace transform to the equation (B.4) with zero initial conditions it is easy to show that the transfer function of the system has the form
011
1
012
21
1
...
...)(
asasas
bsbsbsbsG
nn
n
nn
nn
++++++++= −
−
−−
−− . (B.6)
The transfer function can be expanded in the series
...)( 22
11 ++= −− sgsgsG . (B.7)
From comparison of the right sides of (B.6) and (B.7) we have
...).)(...(
...2
21
1011
1
012
21
1
++++++=
++++−−−
−
−−
−−
sgsgasasas
bsbsbsb
nn
n
nn
nn
(B.8)
Comparing the coefficients at the same powers of s of the equality (B.8) we obtain
....
...,,,
112211
112211
gagagabg
gabgbg
knknknknk
nnn
+−−−−−−
−−−−−−−=
−==
(B.9)
From (B.9) it follows that if the conditions (B.5) are satisfied then +ℜ∈kg
for k = 1,2,…. It is well-known that impulse response g(t) is the original of the transfer
function )]([)( 1 sGtg −= L , where L-1 is the inverse Laplace operator. From (B.7)
B.2 Externally Positive Discrete-Time Linear Systems 495
we have ...!2
)(2
321 +++= tgtggtg . Hence if the conditions (B.5) are satisfied
then +ℜ∈)(tg for all t ≥ 0 and the system described by (B.4) is externally
positive. □
Corollary B.1. The continuous-time system with the transfer function (B.6) is externally positive if the conditions (B.5) are satisfied.
B.2 Externally Positive Discrete-Time Linear Systems
Consider the discrete-time linear system described by the equations
,
,1
iii
iii
DuCxy
BuAxx
+=+=+
(B.10)
where nix ℜ∈ , m
iu ℜ∈ , piy ℜ∈ are the state, input and output vectors,
respectively at the distinct instant +∈ Zi and nnA ×+ℜ∈ , mnB ×
+ℜ∈ ,
npC ×+ℜ∈ , mpD ×
+ℜ∈ .
Definition B.1’. The system (B.10) is called externally positive if for every input
sequence miu +ℜ∈ , +∈ Zi and x0 = 0 the output p
iy +ℜ∈ for all +∈ Zi .
The impulse response gi of single-input single-output system is called the output of the system for the input equal to the unit impulse
≠=
=0
0
for
for
0
1
i
iiδ . (B.11)
In a similar way assuming successively that only one input is equal to δi and the
remaining are zero we may define the matrix of impulse responses mpig ×ℜ∈ of
the system with m-inputs and p-outputs.
Theorem B.3. The system (B.10) is externally positive if and only if its matrix of impulse responses is nonnegative, i.e.
mp
ig ×+ℜ∈ for all +∈ Zi . (B.12)
Proof. The necessity of condition (B.12) follows immediately from Definition B.1’. The output of the system (B.10) with zero initial conditions for any input ui is given by the formula
496 B Positive Regular Continuous-Time and Discrete-Time Linear Systems
=
−=i
kikii ugy
0. (B.13)
If the condition (B.12) is satisfied and miu +ℜ∈ then from (B.13) we have
piy +ℜ∈ for +∈ Zi . □
Theorem B.4. The discrete-time system described by the difference equation (the ARMA model)
=
−=
− =+n
kkik
n
kkiki ubyay
11
(B.14)
is externally positive if
ak ≤ 0 and bk ≥ 0 for k = 1,…, n. (B.15)
Proof. We shall show that if (B.15) holds then +ℜ∈ig for +∈ Zi . Applying the
Z-transform to the equation (B.14) with zero initial conditions it is easy to show that the transfer function of the system has the form
n
n
nn
zazaza
zbzbzbzG −−−
−−−
+++++++=...1
...)(
22
11
22
11 . (B.16)
The transfer function can be expanded in the series
...)( 33
22
11 +++= −−− zgzgzgzG . (B.17)
From comparison of the right sides of (B.16) and (B.17) we have
...).)(...1(
...3
32
21
12
21
1
22
11
+++++++=
+++−−−−−−
−−−
zgzgzgzazaza
zbzbzb
nn
nn
(B.18)
Comparing the coefficients at the same powers of z-1 of the equality (B.18) we obtain
....
...,,,
112211
112211
gagagabg
gabgbg
nkkkk −−− −−−−=−==
(B.19)
From (B.19) it follows that if the conditions (B.15) are satisfied then +ℜ∈ig
for +∈ Zi . Hence by Theorem B.3 the system is externally positive. □
Corollary B.2. The discrete-time system with the transfer function (B.16) is externally positive if the conditions (B.15) are satisfied.
B.3 Internally Positive Continuous-Time Linear Systems 497
B.3 Internally Positive Continuous-Time Linear Systems
Definition B.2. The system (B.1) is called internally positive (shortly positive) if
for any nx +ℜ∈0 and every input mtu +ℜ∈)( we have ntx +ℜ∈)( and pty +ℜ∈)( for all t ≥ 0.
From Definition B.2 it follows that the system (B.1) is internally positive only if its matrix of impulse responses is nonnegative, i.e. the condition (B.2) is satisfied. This condition in general case is not sufficient for the internal positivity of the system (B.1).
The real matrix nnjiaA ×ℜ∈= ][ , is called the Metzler matrix if ai,j ≥ 0 for i ≠
j. Theorem B.5. The continuous-time system (B.1) is internally positive if and
only if the matrix nMA∈ and mnB ×+ℜ∈ , npC ×
+ℜ∈ , mpD ×+ℜ∈ .
Proof. Sufficiency. The solution of the equation (B.1) has the form
−+=t
tAAt dBuexetx0
)(0 )()( τττ . (B.20)
It is well-known that the matrix nnAte ×+ℜ∈ if and only if A is the Metzler
matrix. If A is the Metzler matrix and mnB ×+ℜ∈ , nx +ℜ∈0 , mtu +ℜ∈)( for t ≥ 0
then from (B.20) we obtain ntx +ℜ∈)( for t ≥ 0 and from equation (B.1)
pty +ℜ∈)( since npC ×+ℜ∈ and mpD ×
+ℜ∈ .
Necessity. Let 0)( =tu , t ≥ 0 and iex =0 (the i-th column of the identity matrix
nI ). The trajectory of the system does not leave the orthant n+ℜ only if
,0)0( ≥= iAex what implies 0, ≥jia for .ji ≠ Hence, the matrix A has to be a
Metzler matrix. For the same reason, for 00 =x we have 0)0()0( ≥= Bux what
implies mnB ×+ℜ∈ since mu +ℜ∈)0( may be arbitrary. From (B.1) for
0,0)( ≥= ttu we have 0)0( 0 ≥= Cxy and 0)0( 0 ≥= Cxy and ,npC ×+ℜ∈
since nx +ℜ∈0 may be arbitrary. In a similar way, assuming 00 =x we
obtain 0)0()0( ≥= Duy and ,mpD ×+ℜ∈ since mu +ℜ∈)0( may be
arbitrary. □
The matrix of impulse responses of the system (B.1) is given by the formula
)()( tDBCetg At δ+= for t ≥ 0. (B.21)
498 B Positive Regular Continuous-Time and Discrete-Time Linear Systems
The formula may be obtained by substitution of (B.20) into (B.1) and taking into account that for x0 = 0 and )()( ttu δ= , )()( tgty = . If A is the Metzler
matrix and mnB ×+ℜ∈ , npC ×
+ℜ∈ , mpD ×+ℜ∈ , then from (B.21) it follows that
+ℜ∈)(tg for all t ≥ 0.
We have the following two important corollaries.
Corollary B.3. The matrix of impulse response of internally positive system (B.1) satisfies the condition (B.2).
Corollary B.4. Every continuous-time internally positive system is also externally positive.
B.4 Internally Positive Discrete-Time Linear Systems
Definition B.3. The system (B.10) is called internally positive (shortly positive) if
for any nx +ℜ∈0 and input sequence miu +ℜ∈ , +∈ Zi we have n
ix +ℜ∈ and
piy +ℜ∈ for all +∈ Zi .
From Definition B.3 it follows that the system (B.10) is internally positive only if its matrix of impulse responses is nonnegative, i.e. the condition (B.12) is satisfied. This condition in general case is not sufficient for the internal positivity of the system (B.10).
Theorem B.6. The discrete-time system (B.10) is internally positive if and only if
nnA ×
+ℜ∈ , mnB ×+ℜ∈ , npC ×
+ℜ∈ , mpD ×+ℜ∈ . (B.22)
Proof. Sufficiency. The solution of the equation (B.10) has the form
−
=
−−+=1
0
10
i
kk
kiii BuAxAx . (B.23)
From (B.23) and (B.10) it follows that if (B.22) holds then for nx +ℜ∈0 and
miu +ℜ∈ , +∈ Zi we have n
ix +ℜ∈ and piy +ℜ∈ for all +∈ Zi .
Necessity. Let 0=iu , +∈ Zi . Then from (B.10) for i = 0 we have
nAxx +ℜ∈= 01 and pCxy +ℜ∈= 00 . This implies nnA ×+ℜ∈ and npC ×
+ℜ∈
since nx +ℜ∈0 may be arbitrary. Assuming x0 = 0 from (B.10) for i = 0 we obtain
nBux +ℜ∈= 01 and pDuy +ℜ∈= 00 what implies mnB ×+ℜ∈ and mpD ×
+ℜ∈
since mu +ℜ∈0 may be arbitrary. □
B.5 Asymptotic Stability of Positive Continuous-Time Linear Systems 499
The matrix of impulse responses of the system (B.10) is given by the formula
0
0
for
for1 >
=
= − i
i
BCA
Dg ii +∈ Zi . (B.24)
The formula may be obtained by substitution of (B.23) into (B.10) and taking into account that for x0 = 0 and iiu δ= , ii gy = .
If the conditions (B.22) are satisfied then from (B.24) it follows that +ℜ∈ig
for all +∈ Zi .
Therefore, we have the following two important corollaries.
Corollary B.5. The matrix of impulse response of internally positive system (B.10) satisfies the condition (B.12).
Corollary B.6. Every discrete-time internally positive system is also externally positive.
B.5 Asymptotic Stability of Positive Continuous-Time Linear Systems
The positive system (B.1) is called asymptotically stable if the solution
0)( xetx At= (B.25)
of the equation
nMAxxtAxtx ∈== ,)0(),()( 0 (B.26)
satisfies the condition
0)(lim =
∞→tx
t for every nx +ℜ∈0 . (B.27)
Theorem B.7. The positive system (B.26) is asymptotically stable if and only if one of the following equivalent statements are satisfied:
1) All coefficients of the characteristic polynomial
011
1 ...]det[ asasasAsI nn
nn ++++=− −
− (B.28)
are positive, i.e. 0>ia for i = 1,…,n – 1,
2) All principal minors nii ,...,1, =Δ of the matrix ][ ijaA −=− are
positive, i.e.
0]det[,...,0,02221
12112111 >−=Δ>
−−−−
=Δ>−=Δ Aaa
aaa n
(B.29)
500 B Positive Regular Continuous-Time and Discrete-Time Linear Systems
3) All diagonal entries of the matrices )(kknA − for k = 1,2,…,n – 1 are negative
where the matrices )(kknA − are defined as follows
==
=
=
=−=
−
−−−
−−−
−
−−
−−
)0(,1
)0(,1
)0(1
)0(1,
)0(1,
)0(1
)0(1,1
)0(1,1
)0(1,1
)0(11
)0(1)0(
,)0(1
)0(1
)0(1
)0(,
)0(1,
)0(,1
)0(11
)0(
],...[
,
...
...
...
,
...
...
...
nn
n
nnnnn
nnn
n
nnnn
nn
nnn
n
nn
a
a
baac
aa
aa
Aac
bA
aa
aa
IAA
(B.30)
and
.],...[
,
...
...
...
)(,1
)(,1
)(1
)(1,
)(1,
)(1
)(,
)(1
)(1
)(1
)(,
)(1,
)(,1
)(11
)1(1,1
)1()1()1()(
==
=
=−=
−−−
−
−−−−−−−−
−−−−
−−−−
−−−
−
−+−+−
−−
−−−
−−
kknkn
kkn
kkn
kknkn
kkn
kkn
kknkn
kkn
kkn
kkn
kknkn
kkn
kkn
k
kknkn
kkn
kknk
knk
kn
a
a
baac
ac
bA
aa
aa
a
cbAA
(B.31)
4) The diagonal entries of the lower triangular matrix
=
nnnn aaa
aa
a
A
,2,1,
2221
11
~...~~
0...~~0...0~
~
(B.32)
are negative, i.e.
0~
, <kka for k = 1,…,n. (B.33)
The matrix (B.32) is obtained from the matrix
=
nnnn
n
n
aaa
aaa
aaa
A
,2,1,
,22221
,11211
...
...
...
...
B.6 Asymptotic Stability of Positive Discrete-Time Linear Systems 501
by the use of the row elementary operations. Proof is given in [97, 98].
Theorem B.8. The positive system (B.26) is unstable if at least one diagonal entry of the matrix A is positive, i.e. ak,k > 0 for some ),...,2,1( nk ∈ .
Proof is given in [97, 98].
Definition B.7. A vector xf satisfying the condition Axf = 0 is called the equilibrium point of the continuous-time system (B.26).
Consider the continuous-time single-input (m = 1) system
)()()( tbutAxtx += (B.34)
with a constant positive input 0>u . Let xf be the equilibrium point of the system. Then from (B.34) we have
0=+ ubAx f . (B.35)
Theorem B.9. The equilibrium point xf of asymptotically stable internally positive system (B.34) for 0>u is positive (xf > 0) if b > 0. It is strictly positive (xf >> 0) if b >> 0.
If the internally positive system (B.26) is asymptotically stable then all eigenvalues of the system matrix have negative real parts and det A ≠ 0. In this case the equilibrium point is given by
ubAx f
1−−= . (B.36)
The formula (B.36) follow immediately from (B.35). For asymptotically stable systems we have
ubAtxx
tf
1)(lim −∞→
−==
(B.37)
since 0lim =∞→
At
te .
Corollary B.7. If A is a Metzler matrix of internally positive asymptotically stable
system (B.34) then 01 >− −A .
B.6 Asymptotic Stability of Positive Discrete-Time Linear Systems
The positive system (B.10) is called asymptotically stable if the solution
0i
ix A x= (B.38)
of the equation
502 B Positive Regular Continuous-Time and Discrete-Time Linear Systems
+×
++ ∈ℜ∈= ZiAAxx nnii ,,1 (B.39)
satisfies the condition
lim 0ii
x→∞
= for every nx +ℜ∈0 . (B.40)
Theorem B.10. The positive system (B.39) is asymptotically stable if and only if one of the following equivalent statements are satisfied:
1) Eigenvalues 1 2, , , nz z z of the matrix nnA ×+ℜ∈ have moduli less than
1, i.e. 1kz < for 1, ,k n= ;
2) 0]det[ ≠− AzIn for 1≥z ;
3) 1)( <Aρ (B.41)
where )(Aρ is the spectral radius of the matrix A defined by
{ }knki
zA≤≤
= max)(ρ ;
4) All coefficients ia , 1,...,1,0 −= ni of the characteristic polynomial
011
1 ˆˆ...ˆ])1(det[)( azazazAzIzp nn
nnA ++++=−+= −
− (B.42)
are positive; 5) All principal minors of the matrix
11 12 1
21 22 2
1 2
n
nn
n n nn
a a a
a a aA I A
a a a
= − =
(B.43)
are positive, i.e.
11 1211
21 22
0, 0, ..., det 0a a
a Aa a
> > > , (B.44)
6) There exists a strictly positive vector 0>>x such that
0][ <<− xIA n , (B.45)
7) All diagonal entries of the matrices )(kknA − for k = 1,2,…,n – 1 are negative
where the matrices )(kknA − are defined as follows
B.6 Asymptotic Stability of Positive Discrete-Time Linear Systems 503
==
=
=
=−=
−
−−−
−−−
−
−−
−−
)0(,1
)0(,1
)0(1
)0(1,
)0(1,
)0(1
)0(1,1
)0(1,1
)0(1,1
)0(11
)0(1)0(
,)0(1
)0(1
)0(1
)0(,
)0(1,
)0(,1
)0(11
)0(
],...[
,
...
...
...
,
...
...
...
nn
n
nnnnn
nnn
n
nnnn
nn
nnn
n
nn
a
a
baac
aa
aa
Aac
bA
aa
aa
IAA
(B.46)
and
.],...[
,
...
...
...
)(,1
)(,1
)(1
)(1,
)(1,
)(1
)(,
)(1
)(1
)(1
)(,
)(1,
)(,1
)(11
)1(1,1
)1()1()1()(
==
=
=−=
−−−
−
−−−−−−−−
−−−−
−−−−
−−−
−
−+−+−
−−
−−−
−−
kknkn
kkn
kkn
kknkn
kkn
kkn
kknkn
kkn
kkn
kkn
kknkn
kkn
kkn
k
kknkn
kkn
kknk
knk
kn
a
a
baac
ac
bA
aa
aa
a
cbAA
(B.47)
The proof is given in [97, 98].
Theorem B.11. The positive system (B.39) is unstable if at least one diagonal entry of the matrix A is greater than 1, i.e. ak,k > 1 for some ),...,2,1( nk ∈ .
The proof is given in [97, 98].
Definition B.8. A vector xf satisfying the condition xf = Axf is called the equilibrium point of the discrete-time system (B.39).
Consider the discrete-time single-input (m = 1) system
iii buAxx +=+1 (B.48)
with a constant positive input 0>u . Let xf be the equilibrium point of the system. Then from (B.48) we have
ubAxx ff += . (B.49)
Theorem B.12. The equilibrium point xf of asymptotically stable internally positive system (B.48) for 0>u is positive (xf > 0) if b > 0. It is strictly positive (xf >> 0) if b >> 0.
If the internally positive system (B.48) is asymptotically stable then all eigenvalues of the system matrix have moduli less 1 and det [In – A] ≠ 0. In this case the equilibrium point is given by
504 B Positive Regular Continuous-Time and Discrete-Time Linear Systems
ubAIx nf
1][ −−= . (B.50)
The formula (B.50) follows immediately from equation (B.49). For asymptotically stable systems we have
ubAIxx ni
if
1][lim −∞→
−==
(B.51)
since 0lim 0 =∞→
xAi
t.
Corollary B.8. If A is an asymptotically stable matrix of internally positive system
(B.48) then 0][ 1 >− −AIn . B.7 Asymptotic Stability of Positive Continuous- Time Linear Systems
B.7 Asymptotic Stability of Positive Continuous-Time Linear Systems with Delays in State
B.7 Asymptotic Stability of Positive Continuous- Time Linear Systems
Theorem B.13. The positive continuous-time linear system with q delays in state vector
)()()(
),()()()(1
0
tDutCxty
tBudtxAtxAtxq
kkk
+=
+−+= =
(B.52)
is asymptotically stable if and only if there exists a strictly positive vector n+ℜ∈λ satisfying the condition
=
=<q
kkAAA
0,0λ . (B.53)
Proof. First we shall shown that if the system (B.52) is asymptotically stable then
there exists a strictly positive vector n+ℜ∈λ satisfying (B.53). Integrating the
first equation (B.52) for B = 0 in the interval [0, ∞] we obtain
=
∞∞∞−+=
q
kkk dtdtxAdttxAdttx
1 000
0
)()()(
(B.54)
and
∞
= −=−−∞
01
0)()()0()( dttxAdttxAxx
q
k kdk . (B.55)
B.7 Asymptotic Stability of Positive Continuous-Time Linear Systems 505
For asymptotically stable positive system
0)(,0)()0(,0)(01
0>>+=∞
∞
= −dttxdttxAxx
q
k kdk
(B.56)
and from (B.55) we have (B.53) for ∞
=0
)( dttxλ .
Now we shall show that if (B.53) holds then the positive system (B.52) is asymptotically stable. It is well-known that the system (B.52) is asymptotically stable if and only if the corresponding transpose system
=
−+=q
kk
Tk
T dtxAtxAtx1
0 )()()(
(B.57)
is asymptotically stable. As a candidate for a Lyapunov function for the positive system (B.57) we chose the function
= −
−=q
k
t
kdtk
Tk
T AdxAxxV1
)()( λττλ . (B.58)
which is positive for any nonzero ntx +ℜ∈)( . Using (B.58) and (B.57) we obtain
.)(
))()(()()(
))()(()()(
110
1
λ
λλλ
λλ
Atx
AdtxtxAdtxAtx
AdtxtxtxxV
T
q
kkk
TTq
kkk
TT
q
kkk
TTT
=
−−+−+=
−−+=
==
=
(B.59)
If the condition (B.53) holds then from (B.59) we have 0)( <xV and the system
(B.52) is asymptotically stable.
Remark B.1. As strictly positive vector λ we may choose the equilibrium point
BuAxe1−−= (B.60)
since
0)( 1 <−=−= − BuBuAAAλ for 0>Bu (B.61)
Theorem B.14. The positive continuous-time linear system with delays (B.52) is asymptotically stable if and only if the positive system without delays
506 B Positive Regular Continuous-Time and Discrete-Time Linear Systems
n
q
kk MAAAxx ∈==
=0,
(B.62)
is asymptotically stable.
Proof. The positive system (B.56) is asymptotically stable if and only if there
exists a strictly positive vector n+ℜ∈λ such that (B.53) holds. Hence by Theorem
B.13 the positive system (B.52) is asymptotically stable if and only if the positive system (B.56) is asymptotically stable.
B.8 Reachability of Positive Discrete-Time Linear Systems
Consider the discrete-time (internally) positive linear system described by the equation
iii BuAxx +=+1 , +∈ Zi (B.63)
where nix ℜ∈ , m
iu ℜ∈ are the state and input vectors respectively and
nnA ×+ℜ∈ , mnB ×
+ℜ∈ .
Definition B.9. A state nfx +ℜ∈ of the system (B.63) is called reachable in k
steps if there exists a sequence of inputs miu +ℜ∈ for i = 0,1,…,k – 1 which steers
the zero initial state of the system (x0 = 0) to the state xf.
Theorem B.15. The set of reachable states of positive system (B.63) is a positive convex cone. The cone is solid if and only if the matrix
]...[: 1BAABBR n−= (B.64)
has full row rank n. Proof is given in [101]. The positive system (B.63) is reachable if and only if the reachability cone
is equal to n+ℜ .
Let ei be the i-th (i = 1,2,…,n) column of the identity matrix In. Then aei for a > 0 is called the monomial column.
Definition B.10. A set of all positive linear combinations of columns of positive
matrix nnA ×+ℜ∈ is called its positive image and it will be denoted by Im+A, i.e.
B.9 Reachability of Positive Continuous-Time Linear Systems 507
},:{:Im mn xAxyyA +++ ℜ∈=ℜ∈= . (B.65)
Theorem B.16. The positive system (B.63) is reachable if and only if one of the following equivalent conditions is satisfied:
1) The reachability matrix
]...[ 1BAABBR n−= (B.66)
contains n linearly independent monomial columns,
2) nR ++ ℜ∈Im , where R is defined by (B.64).
Proof is given in [101].
Theorem B.17. The positive system (B.63) is reachable in n steps if and only if it is reachable in (n + 1) steps.
Proof is given in [101].
Corollary B.9. If the positive system (B.63) is reachable, then it is always reachable in n steps.
Proof is given in [101].
Theorem B.18. The positive system (B.63) is reachable if rank R = n and
nmnTT RRR ×
+− ℜ∈1][ . (B.67)
Proof is given in [101].
B.9 Reachability of Positive Continuous-Time Linear Systems
Consider the continuous-time (internally) positive linear system
)()()( tButAxtx += , 0)0( xx = (B.68)
where ntx ℜ∈)( , mtu ℜ∈)( are the state and input vectors respectively and
nnA ×ℜ∈ is a Metzler matrix, mnB ×+ℜ∈ .
Definition B.11. A state nfx +ℜ∈ of the system (B.68) is called reachable in time
tf if there exists an input mtu +ℜ∈)( for ],0[ ftt ∈ which steers the system from
x0 = 0)to the state xf.
Theorem B.19. The positive system (B.68) is reachable if the matrix
508 B Positive Regular Continuous-Time and Discrete-Time Linear Systems
=ft
TATAf deBBeR
0
: τττ for tf > 0 (B.69)
is monomial matrix. The input which steers the state of the system in time tf from x0 = 0 to the state
nfx +ℜ∈ is given by the formula
fftftTAT xReBtu 1)(
)( −−= for ],0[ ftt ∈ . (B.70)
Proof is given in [101]. B.10 Controllability and Controllability to Zero of Pos itive Discrete-Time Linear
B.10 Controllability and Controllability to Zero of Positive Discrete-Time Linear Systems
B.10 Controllability and Controllability to Zero of Pos itive Discrete-Time Linear
Definition B.12. The positive system (B.63) is called controllable in k steps if for
any nonzero initial state nx +ℜ∈0 and a final state nfx +ℜ∈ there exists a
sequence of inputs miu +ℜ∈ for i = 0,1,…,k – 1 which steers the state of the
system from x0 to the state xf.
Definition B.13. The positive system (B.63) is called controllable to zero in k
steps if for any nonzero initial state nx +ℜ∈0 and a final state xf = 0 there exists a
sequence of inputs miu +ℜ∈ for i = 0,1,…,k – 1 which steers the state of the
system from x0 to zero (xf = 0).
Theorem B.20. The positive system (B.63) is controllable in k steps if and only if the matrix A has only zero eigenvalues and one of the conditions of Theorem B.16 is satisfied.
Proof is given in [101].
Theorem B.21. The positive system (B.63) is controllable in infinite number of steps if and only if the system is asymptotically stable and one of the conditions of Theorem B.16 is satisfied.
Proof is given in [101].
Corollary B.10. The controllability of positive system (B.63) implies its reachability.
B.11 Controllability and Controllability to Zero of Positive Continuous-Time Linear Systems
509
Theorem B.22. The positive system (B.63) is controllable to zero:
1) in n steps if and only if matrix A has all zero eigenvalues, 2) in infinite number of steps only if the system is asymptotically stable.
Proof is given in [101].
Corollary B.11. The positive system (B.63) is controllable to zero only if is asymptotically stable.
B.11 Controllability and Controllability to Zero of Positive Continuous-Time Linear Systems
Consider the continuous-time (internally) positive linear system (B.68) with
nMA∈ being a Metzler matrix and mnB ×+ℜ∈ .
Definition B.14. The positive system (B.68) is called controllable if for any
nonzero initial state nx +ℜ∈0 and a final state nfx +ℜ∈ , there exist a time
instant tf > 0 and an inputs mtu +ℜ∈)( for ],0[ ftt ∈ which steers the state of the
system from x0 to the state xf.
Definition B.15. The positive system (B.68) is called controllable to zero if for
any nonzero initial state nx +ℜ∈0 there exist a time instant tf > 0 and an inputs
mtu +ℜ∈)( for ],0[ ftt ∈ which steers the state of the system from x0 to zero
(xf = 0).
Theorem B.23. The positive system (B.68) is controllable if the matrix (B.69) is a monomial matrix for tf > 0 and
nfAt
f xex +ℜ∈− 0 . (B.71)
The input which steers the state of the system in time tf from nx +ℜ∈0 to the state
nfx +ℜ∈ is given by the formula
)()( 0
1)(xexReBtu fAt
fftftTAT −= −−
(B.72)
where the matrix Rf is defined by (B.69). Proof is given in [101].
510 B Positive Regular Continuous-Time and Discrete-Time Linear Systems
Theorem B.24. The positive system (B.68) is not controllable to zero in finite time.
Proof is given in [101].
Theorem B.25. The positive system (B.68) is controllable to zero in infinite time if it is asymptotically stable.
Proof is given in [101].
B.12 Observability of Positive Discrete-Time Linear Systems
Consider the discrete-time (internally) positive linear system (B.10).
Definition B.16. The positive system (B.10) is called observable in k steps if it is
possible to find the unique initial condition nx +ℜ∈0 knowing the system output
in k points y0, y1,…, yk-1 for the initial conditions and zero input ui = 0, +∈ Zi .
Theorem B.26. The positive system (B.10) is observable in n steps if and only if one of the following equivalent conditions is satisfied:
1) The observability matrix
=
−1
:
nCA
CA
C
S
(B.73)
contains n linearly independent monomial rows,
2) nS ++ ℜ=Im .
Proof is given in [101].
Definition B.17. The positive system
iiT
i
iT
iT
i
DuxBy
uCxAx
+=
+=+ ,1
(B.74)
is called the dual system for the positive system (B.10) where the vectors xi, ui, yi and matrices A, B, C, D of both systems are the same.
Theorem B.27. The positive system (B.10) is observable if and only if the dual system (B.74) is reachable.
Proof is given in [101].
Appendix C
Fractional Linear Systems
C.1 Fractional Discrete-Time Linear Systems
The state equations of the fractional discrete-time linear system have the form
kkk
kkk
DuCxy
BuAxx
+=+=Δ +1
α, 10, ≤≤∈ + αZk , (C.1)
where nkx ℜ∈ , m
ku ℜ∈ , pky ℜ∈ are the state, input and output vectors,
respectively and nnA ×ℜ∈ , mnB ×ℜ∈ , npC ×ℜ∈ , mpD ×ℜ∈ .
Definition C.1. The discrete-time function
0
( 1)k
jk k j
j
x xj
α α−
=
Δ = −
, (C.2)
where 10 << α , ℜ∈α and
,...3,2,1
0
for
for
!
)1)...(1(
1
==
+−−=
k
k
k
kk αααα
(C.3)
is called the fractional α order difference of the function xk. Substituting the definition of fractional difference (C.2) into (C.1), we obtain
1
1 11
( 1) ,k
jk k j k k
j
x x Ax Bu k Zj
α+
+ − + +=
+ − = + ∈
(C.4)
or
,)1(
)1(
1
21
1
1
11
11
k
k
jjk
jk
k
k
jjk
jkk
Buxj
xA
Buxj
Axx
+
−+=
+
−+=
+
=+−
+
+
=+−
++
α
α
α
(C.5)
512 C Fractional Linear Systems
where
nIAA αα += . (C.6)
Remark C.1. From (C.5) it follows that the fractional system is equivalent to the system with increasing number of delays.
In practice it is assumed that j is bounded by natural number h. In this case the equations (C.1) take the form
.
,1
)1(1
1
kkk
k
h
jjk
jkk
DuCxy
ZkBuxj
xAx
+=
∈+
+
−+= +=
−+ α
α
(C.7)
Theorem C.1. The solution of the equation (C.4) has the form
−
=−−Φ+Φ=
1
010
k
iiikkk Buxx , (C.8)
where the matrices kΦ are determined by the equation
.
,)1()(
0
1
21
11
n
k
iik
inkk
I
iIA
=Φ
Φ
−++Φ=Φ
+
=+−
++
αα
(C.9)
Proof is given in [168].
Definition C.2. The system (C.4) is called the (internally) positive fractional
system if nkx +∈ ℜ and ,p
ky +∈ ℜ k Z+∈ for every initial conditions 0nx +∈ℜ
and all input sequences ,mku +∈ ℜ .k Z+∈
Using (C.3) it is easy to show that if 10 << α , then
,...2,1,0)1( 1 =>
− + i
ii α
. (C.10)
Theorem C.2. The fractional system (C.4) is (internally) positive if and only if
nnnIAA ×
+ℜ∈+= ][ αα , ,n mB ×+∈ℜ ,p nC ×
+∈ ℜ p mD ×+∈ℜ . (C.11)
Proof is given in [168].
Definition C.3. The fractional discrete-time system (C.4) is called externally
positive system if for any input sequences ,mku +∈ ℜ k Z+∈ and x0 = 0
we have ,pky +∈ ℜ k Z+∈ .
C.2 Reachability of Fractional Discrete-Time Linear Systems 513
Definition C.4. The output of single-input single-output (SISO) linear system for unite impulse
>=
=Δ=0
0
for
for
0
1
i
iu ii
(C.12)
and zero initial conditions is called the impulse response of the system.
Theorem C.3. The fractional discrete-time system (C.4) is externally positive if and only if its impulse response matrix
==
Φ=
− ,...2,1
0
for
for
1 k
k
BC
Dg
kk
(C.13)
is nonnegative, i.e. p m
kg ×+∈ ℜ for k Z+∈ . (C.14)
Proof is given in [168].
C.2 Reachability of Fractional Discrete-Time Linear Systems
Definition C.5. A state nfx ℜ∈ is called reachable in (given) q steps if there
exists an input sequence mku ℜ∈ , 0,1, , 1k q= − which steers the state of the
system (C.4) from zero )0( 0 =x to the final state fx , i.e. q fx x= . If every
given nfx ℜ∈ is reachable in q steps then the system (C.4) is called reachable in
q steps. If for every nfx ℜ∈ there exists a number q of steps such that the
system is reachable in q steps then the system is called reachable.
Theorem C.4. The fractional system (C.4) is reachable in q steps if and only if
nBBB q =ΦΦ − ]...[rank 11 . (C.15)
Theorem C.5. In the condition (C.15) the matrices 11,..., −ΦΦ q can be
substituted by the matrices 1,..., −qAA αα i.e.
nBABABBBB qq ==ΦΦ −
− ]...[rank]...[rank 111 αα . (C.16)
Proof is given in [168].
Theorem C.6. The fractional system (C.4) is reachable if and only if one of the equivalent conditions is satisfied:
1) The matrix ][ BAzIn α− has full rank i.e.
514 C Fractional Linear Systems
CznBAzIn ∈∀=− ,][rank α , (C.17)
2) The matrices ][ αAzIn − , B are relatively left prime or equivalently it is
possible using elementary column operations (R) to reduce the matrix ][ BAzIn α− to the form ]0[ nI i.e.
]0[][ nR
n IBAzI ⎯→⎯− α . (C.18)
Proof is given in [168].
C.3 Reachability of Positive Fractional Discrete-Time Linear Systems
Definition C.6. A state nfx +ℜ∈ of the positive fractional system (C.4) is called
reachable in q steps if there exists an input sequence mku +ℜ∈ , 0,1, , 1k q= −
which steers the state from zero )0( 0 =x to the final state fx , i.e. q fx x= . If
every given nfx +ℜ∈ is reachable in q steps then the positive system (C.4) is
called reachable in q steps. If for every nfx +ℜ∈ there exists a number q of steps
such that the system is reachable in q steps then the system (C.4) is called reachable.
Definition C.7. A square real matrix is called monomial if its every column and its every row has only one positive entry and the remaining entries are zero.
Theorem C.7. The positive fractional system (C.4) is reachable in q steps if and only if
]...[ 11 BBBR qq −ΦΦ= C.19)
contains n linearly independent monomial columns. Proof is given in [168].
Theorem C.8. The positive fractional system (C.4) is reachable only if the matrix
],[ BIA nα+ (C.20)
contains at least n linearly independent monomial columns. Proof is given in [168].
C.4 Controllability to Zero of Fractional Discrete-Time Linear Systems 515
C.4 Controllability to Zero of Fractional Discrete-Time Linear Systems
Definition C.8. The fractional system (C.4) is called controllable to zero in (given)
q steps if there exists an input sequence miu ℜ∈ , 1,...,1,0 −= qi which steers
the state of the system from 0x ≠ 0 to the final state .0=fx The fractional
system (C.4) is called controllable to zero if there exists a natural number q such that the system is controllable to zero in q steps.
Theorem C.9. The fractional system (C.4) is controllable to zero in q steps if
nBBB q =ΦΦ − ]...[rank 11 . (C.21)
Proof is given in [168].
Theorem C.10. For the controllability to zero of the fractional system (C.4) the following equality holds
]...[rank]...[rank 111 BABABBBB q
q−
− =ΦΦ αα . (C.22)
Proof is given in [168].
Theorem C.11. The fractional system (C.4) is controllable to zero if and only if
]...[rank]...[rank 11 BABABBABAB qq
q −− =Φ αααα . (C.23)
Proof is given in [168].
Remark C.2. The condition (C.21) is only sufficient condition but not necessary for the controllability to zero of the system (C.4) since condition (C.21) implies only the condition (C.23).
Theorem C.12. The fractional system (C.4) is controllable to zero if and only if one of the equivalent conditions is satisfied:
1) The matrix ][ BdAIn α− has full rank i.e.
CdnBdAIn ∈∀=− ,][rank α , (C.24)
2) The matrices ][ dAIn α− , B are relatively left prime or equivalently
it is possible using elementary column operations (R) to reduce the matrix ][ BdAIn α− to the form ]0[ nI i.e.
]0[][ nR
n IBdAI ⎯→⎯− α . (C.25)
Proof is given in [168].
516 C Fractional Linear Systems
C.5 Controllability to Zero of Positive Fractional Discrete-Time Linear Systems
Definition C.9. The positive fractional system (C.4) is called controllable to zero
in q steps if for any nonzero initial state nx +ℜ∈0 there exists an input sequence
miu +ℜ∈ , 1,...,1,0 −= qi which steers the state of the system from 0x to zero
).0( =fx The positive fractional system (C.4) is called controllable to zero if
there exists a natural number q > 0 such that the system is controllable to zero in q steps.
Theorem C.13. The positive fractional system (C.4) with 0≠B is controllable to zero in q steps if and only if
0=Φq . (C.26)
Moreover 0=iu for .1,...,1,0 −= qi
Proof is given in [168].
C.6 Practical and Asymptotic Stability of Fractional Discrete-Time Linear Systems
In practical problems it is assumed that number of delays is bounded by some natural number h. In that case the equation (C.5) takes the form
.,
,1
1
+
=−+
∈+=
++=
ZkDuCxy
BuxcxAx
kkk
k
h
jjkjkk α
(C.27)
Definition C.10. The positive fractional system (C.1) is called practically stable if and only if the system (C.27) with bounded to h number of delays is asymptotically stable.
Defining the new state vector
1
k
kk
k h
x
xx
x
−
−
=
(C.28)
we may write the equations (C.27) in the form
1 ,k k kx Ax Bu k Z+ += + ∈ ,
k k ky Cx Du= + , (C.29)
C.6 Practical and Asymptotic Stability of Fractional Discrete-Time Linear Systems 517
where
nhnDDCC
B
B
I
I
I
IcIcIcA
A
mpnp
mnnn
n
n
n
nhnhn
)1(~,~
,]0...0[~
,
0
0~,
0...00
00...0
00...0
...
~
~
~~~
11
+=ℜ∈=ℜ∈=
ℜ∈
=ℜ∈
=
×+
×+
×+
×+
−
α
(C.30)
where
−== +
jcc j
jjα
α 1)1()( for ,...2,1=j
To test the practical stability of the positive fractional system (C.1) the following theorem can be used to the system (C.29).
Theorem C.14. The positive fractional system (C.1) is practically stable if and only if one of the following equivalent conditions is satisfied:
1) Eigenvalues , 1,...,kz k n= of the matrix A~
have moduli less than 1, i.e.
| | 1kz < for 1,...,k n= , (C.31)
2) 0]~
det[ ~ ≠− AzIn for 1≥z ,
3) 1)~
( <Aρ where )~
(Aρ is the spectral radius of the matrix A~
defined by
1( ) max{| |}k
k nA zρ
≤ ≤=
,
4) All coefficients , 0,1,..., 1ia i n= − of the characteristic polynomial
011
1~ ~~...~]~
)1(det[)( azazazAzIzp nn
nnA ++++=−+= −
− (C.32)
of the matrix [ ]nA I− are positive,
5) All leading principal minors of the matrix
11 12 1
21 21 2
1 1
...
...[ ]
... ... ... ...
...
n
nn
n n nn
a a a
a a aI A
a a a
− =
(C.33)
are positive , i.e.
11 1211
21 22
| | 0, 0 ,..., det[ ] 0n
a aa I A
a a> > − >
(C.34)
518 C Fractional Linear Systems
6) There exist strictly positive vectors ,nix +∈ ℜ 0,1,...,i h= satisfying
0 1 1 2 1, ,..., h hx x x x x x−< < < (C.35)
such that
0 1 1 0... h hA x c x c x xα + + + < . (C.36)
Proof is given in [168].
Definition C.11. The positive fractional system (C.1) is called asymptotically stable if the system is practically stable for ∞→h .
Lemma C.1. If 10 << α then
∞
=−=
11
jjc α , (C.37)
where the coefficients jc are defined by
−== +
jcc j
jjα
α 1)1()( for ,...2,1=j .
Proof is given in [168].
Theorem C.15. The positive fractional system (C.1) is asymptotically stable if and only if positive system
ini xIAx )(1 +=+ (C.38)
is asymptotically stable. Proof is given in [168]. Applying to the positive system (C.38) Theorem C.15 we obtain the following
theorem.
Theorem C.16. The positive fractional system (C.1) is asymptotically stable if and only if one of the equivalent conditions holds:
1) Eigenvalues kzzz ,...,, 21 of the matrix nIA + have moduli less than 1,
i.e. 1<kz for nk ,...,1= ,
2) All coefficients of the characteristic polynomial of the matrix A are positive,
3) All leading principal minors of the matrix A− are positive.
Theorem C.17. The positive fractional system (C.1) is unstable if at least one diagonal entry of the matrix A is positive.
C.7 Fractional Different Orders Discrete-Time Linear Systems
Consider the fractional different orders discrete-time linear system [168]
),()()()1(
),()()()1(
22221212
12121111
kuBkxAkxAkx
kuBkxAkxAkx
++=+Δ
++=+Δβ
α
+∈ Zk , (C.39)
C.7 Fractional Different Orders Discrete-Time Linear Systems 519
where 11 )( nkx ℜ∈ and 2
2 )( nkx ℜ∈ are the state vectors, jninijA
×ℜ∈ ,
miniB ×ℜ∈ ; i, j = 1,2, and mku ℜ∈)( is the input vector.
The fractional derivative of α order is defined by
,....2,1,1)0(
,!
)1)...(1()1()1()(
),()()()1()(00
==
+−−−=
−=
−=−
−=Δ
==
jc
j
j
jjc
jkxjcjkxj
kx
jj
k
j
k
j
j
α
α
αα
αααα
α
(C.40)
Using (C.40) we can write the equation (C.39) in the form
),()1()()()()1(
),()1()()()()1(
2
1
22221212
1
1
21212111
kuBjkxjckxAkxAkx
kuBjkxjckxAkxAkx
k
j
k
j
++−−+=+
++−−+=+
+
=
+
=
ββ
αα
(C.41)
where 1111 nIAA αα += , 2221 nIAA ββ += .
Theorem C.18. The solution to the fractional equation (C.39) with initial conditions 101 )0( xx = , 202 )0( xx = is given by
−
=−−
Φ+
Φ=
1
0 2
11
20
10
2
1 )()(
)( k
iikk iu
B
B
x
x
kx
kx, (C.42)
where kΦ is defined by
++==
=
Φ−−Φ−ΦΦ−−Φ−Φ
+==Φ
−−−−
−−−,...2,1
,...,2,1
0
for
for
for
...
...
)(
1211
01211
21
lki
ki
i
DDA
DDA
nnnI
kikii
iii
n
i
(C.43)
Proof. From definition of the inverse matrix we have
nk
kn IzzzzDzDzDAzI =+Φ+Φ+Φ−−−−− −−−−−− ...]][...[ 32
21
10
22
11 , (C.44)
where
+
+=
=
2
1
221
121
)1(0
0)1(,
n
nk Ikc
IkcD
AA
AAA
β
α
β
α. (C.45)
Comparing the coefficients at the same power of z-1 we obtain
520 C Fractional Linear Systems
,...,,, 0211230112010 Φ−Φ−Φ=ΦΦ−Φ=ΦΦ=Φ=Φ DDADAAIn . (C.46)
□
C.8 Positive Fractional Different Orders Discrete-Time Linear Systems
Consider the fractional different orders discrete-time linear systems described by the equation (C.39) and
)()(
)()(
2
1 kDukx
kxCky +
= , (C.47)
where 11 )( nkx ℜ∈ , 2
2 )( nkx ℜ∈ , mku ℜ∈)( , pky ℜ∈)( are the state, input
and output vectors and npC ×ℜ∈ , mpD ×ℜ∈ .
Definition C.12. The fractional system (C.39), (C.47) is called positive if and only
if 11 )( nkx +ℜ∈ , 2
2 )( nkx +ℜ∈ and pky +ℜ∈)( , +∈ Zk for any initial
conditions 1101 )0( nxx +ℜ∈= , 2
202 )0( nxx +ℜ∈= and all input sequences
mku +ℜ∈)( , +∈ Zk .
Theorem C.19. The fractional discrete-time linear system (C.39), (C.47) with 10 << α , 10 << β is positive if and only if
nn
AA
AAA ×
+ℜ∈
=
β
α
221
121 , mn
B
BB ×
+ℜ∈
=
2
1 , npC ×+ℜ∈ , mpD ×
+ℜ∈ . (C.48)
Proof is given in [168]. These considerations can be easy extended to fractional systems consisting
of n subsystems of different fractional order [168].
C.9 Fractional Continuous-Time Linear Systems
Consider the continuous-time fractional linear system described by the state equations
),()()(
,10),()()(0
tDutCxty
tButAxtxDt
+=≤<+= αα
(C.49)
where ,)( Ntx ℜ∈ ,)( mtu ℜ∈ pty ℜ∈)( are the state, input and output vectors
and ,NNA ×ℜ∈ ,mNB ×ℜ∈ ,NpC ×ℜ∈ .mpD ×ℜ∈
C.9 Fractional Continuous-Time Linear Systems 521
The following Caputo definition of the fractional derivative will be used
,)(
)(
)(
1)()(
01
)(
0 −+−−Γ==
t
n
n
t dt
f
ntf
dt
dtfD τ
ττ
α αα
αα
,...}2,1{1 =∈≤<− Nnn α , (C.50)
where ℜ∈α is the order of fractional derivative, n
nn
d
fdf
τττ )(
)()( =
and ∞
−−=Γ0
1)( dttex xt is the gamma function.
Theorem C.20. The solution of equation (C.49) is given by
,)0(,)()()()(0
000 =−Φ+Φ=t
xxdButxttx τττ
(C.51)
where
,)1(
)()(0
0 ∞
= +Γ==Φ
k
kk
k
tAAtEt
α
αα
α
(C.52)
∞
=
−+
+Γ=Φ
0
1)1(
])1[()(
k
kk
k
tAt
α
α
(C.53)
and )( αα AtE is the Mittage-Leffler matrix function.
Proof is given in [168].
Remark C.3. From (C.52) and (C.53) for 1=α we have
∞
==
+Γ=Φ=Φ
00 )1(
)()()(
k
Atk
ek
Attt .
Remark C.4. From the classical Cayley-Hamilton theorem we have the following. If
011
1 ...)()(]det[ asasasAsI NN
NN ++++=− −
−αααα
(C.54)
then
0... 011
1 =++++ −− IaAaAaA N
NN . (C.55)
522 C Fractional Linear Systems
C.10 Positivity of Fractional Continuous-Time Linear Systems
Definition C.13. The fractional system (C.49) is called the internally positive
fractional system if and only if Ntx +ℜ∈)( and pty +ℜ∈)( for 0≥t for any
initial conditions Nx +ℜ∈0 and all inputs ,)( mtu +ℜ∈ .0≥t
Definition C.14. A square real matrix ][ ijaA = is called the Metzler matrix
if its off-diagonal entries are nonnegative, i.e. 0≥ija for ji ≠ .
Let NNA ×ℜ∈ and 10 ≤< α . Then
0for)1(
)(0
0 ≥ℜ∈+Γ
=Φ ×+
∞
= t
k
tAt NN
k
kk
α
α
(C.56)
and
0for])1[(
)(0
1)1(≥ℜ∈
+Γ=Φ ×
+∞
=
−+ t
k
tAt NN
k
kk
α
α
(C.57)
if and only if A is a Metzler matrix. Proof is given in [168].
Definition C.15. The fractional system (C.49) is called externally positive if
0,)( ≥ℜ∈ + tty p for every input 0,)( ≥ℜ∈ + ttu m and .00 =x
Definition C.16. The impulse response )(tg of single-input single-output system
is called its output for the input equal to the Dirac impulse )(tδ with zero initial
conditions. Assuming successively that only one input is equal to )(tδ and the
remaining inputs and initial conditions are zero we may define the impulse
response matrix mptg ×ℜ∈)( of the system (C.49).
The impulse response matrix of the system (C.49) is given by
0for)()()( ≥+Φ= ttDBtCtg δ . (C.58)
Proof is given in [168].
Theorem C.21. The continuous-time fractional system (C.49) is externally positive if and only if
0for)( ≥ℜ∈ ×+ ttg mp . (C.59)
Proof is given in [168].
Corollary C.1. The impulse response matrix (C.58) of the internally positive system (C.49) is nonnegative.
C.11 Reachability of Fractional Positive Continuous-Time Linear Systems 523
Corollary C.2. Every continuous-time fractional internally positive system (C.49) is also externally positive.
C.11 Reachability of Fractional Positive Continuous-Time Linear Systems
Definition C.17. The state Nf Rx +∈ of the fractional positive system (C.49) is
called reachable in time ft if there exist an input ],0[,)( fm tttu ∈ℜ∈ + which
steers the state of system (C.49) from zero initial state 00 =x to the state .fx If
every state Nf Rx +∈ is reachable in time ft the system is called reachable in
time ft . If for every state Nf Rx +∈ there exist a time ft such that the state is
reachable in time ft then the system (C.49) is called reachable.
Theorem C.22. The continuous-time fractional system (C.49) is reachable in time
ft if the matrix
ΦΦ=t
TTf dBBtR
0
)()()( τττ
(C.60)
is a monomial matrix. The input which steers the state of the system (C.49)
from 00 =x to fx is given by the formula
fffTT xtRttBtu )()()( 1−−Φ= . (C.61)
Proof is given in [168].
Theorem C.23. If matrix NNNaaaA ×
+ℜ∈= ],...,,[blockdiag 21 and mNB ×+ℜ∈
for m = N is monomial matrix then the continuous-time fractional system (C.49) is reachable.
Proof is given in [168].
Appendix D
Minimal and Cyclic Realizations and Structure Stability of Normal Transfer Matrices
D.1 Basic Notions and Problem Formulation
Following [100] let us consider the linear continuous-time system
),()()(
),()()(
tDutCxty
tButAxtx
+=+=
(D.1)
where ntx ℜ∈)( is the state vector, mtu ℜ∈)( is the input vector and
pty ℜ∈)( is the output vector and nnA ×ℜ∈ , mnB ×ℜ∈ , npC ×ℜ∈ ,
mpD ×ℜ∈ . The transfer matrix of the system (D.1) is given by
)(][)( 1 sDBAsICsT mpn
×− ℜ∈+−= . (D.2)
For the given matrices A, B, C, and D there is only one unique T(s), but for given T(s) there exist many different A, B, C, D satisfying (D.2).
Definition D.1. Matrices A, B, C, D satisfying (D.2) are called a realization of a given transfer matrix T(s).
Definition D.2. A realization is called minimal if the matrix A has the smallest size among all realizations of T(s).
A matrix nnA ×ℜ∈ is called a cyclic matrix if its minimal polynomial Ψ(s) coincides with its characteristic polynomial Ψ(s) = Φ(s) = det[Ins – A].
Definition D.3. A minimal realization is called cyclic (or simple) if the matrix A is cyclic.
The matrix D for a given proper transfer matrix T(s) can be computed using the formula
)(lim sTDs ∞→
=
(D.3)
which results from (D.2), since
526 D Minimal and Cyclic Realizations and Structure Stability
0][lim 1 =− −∞→
AsIns
. (D.4)
From (D.2) and (D.3) we have
BAsICDsTsT nsp1][)()( −−=−= . (D.5)
The realization problem can be stated as follows: Given a proper transfer matrix T(s), find a realization of T(s).
The problem for cyclic realizations can be stated as follows: Given a proper rational transfer matrix T(s), find a cyclic realization of T(s).
D.2 Existence of the Minimal and Cyclic Realizations
D.2.1 Existence of the Minimal Realizations
The following theorem formulates necessary and sufficient conditions for the existence of a minimal realization for a given rational proper transfer matrix T(s).
Theorem D.1. A realization (A, B, C, D) of the transfer matrix T(s) is minimal if and only if the pair (A, B) is controllable and the pair (A, C) is observable.
Proof is given in [100].
Theorem D.2. If (A, B, C, D) and ),,,( DCBA are two minimal realizations
of the transfer matrix T(s) then there exists a unique nonsingular matrix P such that
CPCBPBAPPA === −− ,, 11 (D.6)
and the matrix P is given by
111 ]][[][ −−− == HHHHRRRRP TTTT , (D.7)
where
.,
],...[],...[
11
11
=
=
==
−−
−−
nn
nn
AC
AC
C
H
CA
CA
C
H
BABABRBAABBR
(D.8)
Proof is given in [100].
D.2 Existence of the Minimal and Cyclic Realizations 527
D.2.2 Existence of the Cyclic Realizations
Rational matrix T(s) = N(s)/d(s) is called normal if every nonzero second-order minor of the polynomial matrix N(s) is divisible (without reminder) by the polynomial d(s).
The following theorem formulates necessary and sufficient conditions for the existence of a cyclic realization for a given rational proper transfer matrix T(s).
Theorem D.3. If A is a cyclic matrix and the pair (A, B) is controllable, then
]det[
][)(
AsI
BAsIsW
n
adn
−−=
(D.9)
is an irreducible and normal matrix. If A is a cyclic matrix and the pair (A, C) is observable, then
]det[
][)(
AsI
AsICsW
n
adn
−−=
(D.10)
is an irreducible and normal matrix.
Theorem D.4.The rational matrix
)(]det[
][)( s
AsI
BAsICsW mp
n
adn ×ℜ∈−
−=
(D.11)
is irreducible if and only if the matrices A, B and C are a cyclic realization of the matrix W(s).
Proof. Necessity. If the matrices A, B and C do are not a cyclic realization, then A is not a cyclic matrix or pair (A, B) is not controllable or pair (A, C) is not observable. If A is not cyclic then
]det[
][][ 1
AsI
AsIAsI
n
adnn −
−=− −
(D.12)
is a reducible matrix. If (A, B) is not controllable pair, then
]det[
][
AsI
BAsI
n
adn
−−
(D.13)
is a reducible matrix and if (A, C) is not observable pair, then
]det[
][
AsI
AsIC
n
adn
−−
(D.14)
is reducible as well.
528 D Minimal and Cyclic Realizations and Structure Stability
Sufficiency. According to Theorem D.3, if A is a cyclic matrix and the pair (A, B) is controllable, then matrix (D.9) is an irreducible and if the pair (A, C) is observable, then the matrix (D.10) is an irreducible. Thus if the matrices A, B and C are a cyclic realization, then the matrix (D.11) is irreducible. □
Theorem D.5. There exists a cyclic realization for a rational proper (transfer) matrix T(s) if and only if T(s) is a normal matrix.
Proof. Necessity. If there exists a cyclic realization (A, B, C, D) of T(s) then [In – A]-1 is a normal matrix and according to Binet-Cauchy Theorem [In – A]-1B is normal matrix. Normality of the matrix C[In – A]-1 follows by virtue of Theorem D.3.
Sufficiency. If
)()(
)()( s
sm
sLsT mp×ℜ∈=
(D.15)
is a normal matrix, then using (D.3) we can compute matrix D and the strictly proper matrix (D.5), and in turn compute the cyclic matrix A with the dimensions n×n, n = deg m(s), the controllable pair (A, B) and observable pair (A, C). □ D.3 Computation of Cyclic Realizations wit h the Matrix A
D.3 Computation of Cyclic Realizations with the Matrix A in the Frobenius Canonical Form
D.3 Computation of Cyclic Realizations wit h the Matrix A
The problem of computation a cyclic realization (AF, B, C, D) for rational matrix T(s), with the matrix AF in Frobenius form, can be stated as follows: Given a rational proper matrix T(s) find a minimal realization with the matrix AF in Frobenius form
−−−−
=
−1210 ...
1...000
0...100
0...010
n
F
aaaa
A . (D.16)
Given T(s) and using (D.3) we can compute the matrix D, and the strictly proper rational matrix
)()(
)(][)()( 1 s
sm
sLBAsICDsTsT mp
Fnsp×− ℜ∈=−=−= . (D.17)
Thus the problem is reduced to computing a minimal realization of the strictly proper matrix Tsp(s).
The characteristic polynomial m(s) of the matrix (D.16), which is equal to the minimal polynomial Ψ(s), has the form
D.3 Computation of Cyclic Realizations with the Matrix A 529
011
1 ...]det[)()( asasasAsIssm nn
nn ++++=−=Ψ= −
− . (D.18)
It is easy to show that [Ins – AF]ad of the matrix (D.16) has the form
][)()(
1)(
...
1...000
0...10
0...01
][
1210
ssksM
sw
asaaa
s
s
AsI nn
adn
adFn×
−
ℜ∈
=
+−
−−
=− ,
(D.19) where
.)(
...
,...)(
,...)(
,]...[)(
],)(...)([)(
11
233
12
2
122
11
1
12
11
−
−−
−−
−−
−−
−−
+=
++++=
++++=
=
=
n
nn
nn
nn
nn
Tn
n
assm
asasassm
asasassm
ssssk
smsmsw
(D.20)
In order to perform the structural decomposition [100] of the inverse [Ins – AF]-1, we reduce the matrix (D.5) to the form (D.19). To this end we premultiply the matrix (D.5) by
−
=−
−
1
1,1
)(
01)(
n
n
IsksU (D.21)
and postmultiply it by the unimodular matrix
−
= −−)(1
0)( 11,1
sw
IsV nn . (D.22)
Now we obtain
−
=−−
−)()()(0
01)(])[(
1,1
1,1
swsksMsVAsIsU
n
nadFn , (D.23)
1][ −− Fn AsI is a normal matrix. Every nonzero second-order minor is divisible
without remainder by m(s). Thus every entry of )()()()( swsksMsM −=
is divisible without reminder by m(s). Therefore we have
530 D Minimal and Cyclic Realizations and Structure Stability
)(ˆ)()( sMsmsM = , ][)(ˆ )1()1( ssM nn −×−ℜ∈ . (D.24)
Taking into account that
=
−
−−
1
1,11
)(
01)(
n
n
IsksU ,
=
−−
−
1,11
10
1)()(
nnI
swsV (D.25)
as well as (D.24) and (D.23) we obtain
)()()()()()(ˆ)(0
01)(][ 1
1,1
1,11 sGsmsQsPsVsMsm
sUAsI FFFn
nadFn +=
=− −
−
−− ,
(D.26)
where
.0)(ˆ
00)(
)(ˆ0
00)()(
],1)([)(]01[)(
,)(
1
0
1)()(
1,1
1,11
1,1
1,11
11,1
1,1
1
=
=
==
=
=
−
−−
−
−−
−−
−
−
n
n
n
nF
nF
nF
sMsV
sMsUsG
swsVsQ
sksUsP
(D.27)
From (D.17) and (D.26)we have
)()()()()()()()(][)( sGsmsQsPBsCGsmBsQsCPBAsICsL FFFadFn +=+=−= ,
(D.28)
where
.0)(ˆ
00)()(
,]1)([)()(
,)(
1)()(
1,1
1,1B
sMCBsCGsG
BswBsQsQ
skCsCPsP
n
nF
F
F
==
==
==
−
−
(D.29)
Let Ci be the i-th column of the matrix C, and Bi the i-th row of the matrix B, i = 1,2,…,n.
D.3 Computation of Cyclic Realizations with the Matrix A 531
Taking into account (D.29) and (D.20) we obtain
,...
)...(...)()(
)(...)(]1)(...)([)(
,......
1
]...[)(
121
22113
321122
2111
1
1111
1
11
211
21
1
1
−
−−−
−−
−
−−−
−
−
+++=
++++++++++=
+++=
=
+++=+++=
=
nn
nn
nnn
nn
nnn
n
n
nn
nn
n
n
sQsQQ
BBaBasBBaBasBBasB
BsmBsmB
B
B
smsmsQ
sPsPPsCsCC
s
sCCsP
(D.30)
where
ii CP = for i = 1,2,…,n, (D.31)
....,...,, 112211121111 nnnnnn BBaBaBaQBBaQBQ ++++=+== −−−− (D.32)
With Qi, i = 1,2,…,n known we can recursively compute from (D.32) the rows Bi, i = 1,2,…,n of the matrix B
....,...,, 112211111121 −−−− −−−−=−== nnnnnn BaBaBaQBBaQBQB (D.33)
From the above considerations we have the following procedure for computing the desired cyclic realization (AF, B, C, D) of a given transfer matrix T(s).
Procedure D.1
Step 1. Using (D.3), compute the matrix D and the strictly proper matrix (D.17). Step 2. With the coefficients ai, i = 1,2,…,n – 1 of the known polynomial m(s),
compute the matrix AF given by (D.16). Step 3. Performing the decomposition of the polynomial matrix L(s), compute
the matrices P(s) and Q(s). Step 4. Using (D.31) and (D.33), compute the matrices C and B.
Example D.1. Using Procedure D.1, compute the cyclic realization of the rational matrix
+++++++++−−
+++=
25222
221
12
1)(
2323
233
23 ssssss
sssss
ssssT . (D.34)
It is easy to check that the matrix (D.34) is normal. Thus there exists its cyclic realization.
Using Procedure D.1 we compute.
532 D Minimal and Cyclic Realizations and Structure Stability
Step 1. Using (D.3) and (D.17) we obtain
−==
∞→ 21
11)(lim sTD
s (D.35)
and
−++
+++=−=
s
ss
sssDsTsTsp
1
12
12
1)()(
2
23. (D.36)
Step 2. In this case a0 = 1, a1 = 2, a2 = 1 and
−−−=
121
100
010
FA . (D.37)
Step 3. In order to perform the structural decomposition of the matrix
−++=
s
sssL
1
12)(
2
(D.38)
is sufficient to interchange its columns, i.e. to postmultiply it by
=
01
10)(sV
(D.39)
and compute P(s) and Q(s)
−−−−
+++
=
−++=
−++=
120
00]21[
1
1
2101
10
1
12)()(
232
22
sssss
s
s
ss
s
sssVsL
(D.40)
that is
]12[01
10]21[)(,
1)( 22 ++=
++=
= sssssQ
ssP . (D.41)
Step 4. Taking into account that
ssPPsP
+
=+=
1
0
0
1)( 21
(D.42)
D.4 Computation of Cyclic Realizations with the Matrix A 533
and
22
321 ]01[]01[]12[)( sssQsQQsQ ++=++= (D.43)
from (D.31) and (D.33) we obtain
].10[]00[1]01[2]12[
],00[]01[1]01[],01[
,0
0,
1
0,
0
1
221113
122231
332211
=−−=−−==−=−===
==
==
==
BaBaQB
BaQBQB
PCPCPC
(D.44)
Hence the desired matrices B and C are
=
=
=
010
001][,
10
00
01
321
3
2
1
CCCC
B
B
B
B . (D.45)
It is easy to check that (AF, B) (determined by (D.37) and (D.45)) is controllable pair and (AF, C) is and observable pair. Thus the obtained realization is cyclic.
D.4 Computation of Cyclic Realizations wit h the Matrix A D.4 Computation of Cyclic Realizations with the Matrix A
in the Jordan Canonical Form D.4 Computation of Cyclic Realizations wit h the Matrix A
The problem of computation a cyclic realization (AJ, B, C, D) for rational matrix T(s), with the matrix AJ in Jordan canonical form, can be stated as follows: Given a rational proper matrix T(s) find a minimal realization with the matrix AJ in Jordan canonical form
],...,,[ blockdiag
...0
0...
21
1
p
p
J JJJ
J
J
A =
=
(D.46)
with
imim
i
i
i
i
iimim
i
i
i
i
i
s
s
s
s
J
s
s
s
s
J ×× ℜ∈
=ℜ∈
=
1...00
0...00
00...1
00...0
'or
0...00
1...00
00...0
00...1
,
(D.47)
where i = 1,2,…,p and si are different poles with multiplicities mi respectively,
534 D Minimal and Cyclic Realizations and Structure Stability
=
=p
ii nm
1
(D.48)
of the matrix T(s). With the matrix T(s) given, and using (D.3) we compute the matrix D
and then the strictly proper rational matrix (D.5). The problem has been reduced to computing a minimal realization of the
strictly proper matrix Tsp(s). At the beginning consider the case of multiplicity one (mi = 1, i = 1,2,…,p)
of the matrix
)(
)()(
sm
sLsTsp = , (D.49)
where
njiiisssssssssm jin ,...,2,1,,,),)...()(()( 21 =≠≠−−−= . (D.50)
and si = 1, i = 1,2,…,n are real numbers. In this case Tsp(s) can be expressed in the following form
= −
=n
i i
isp ss
TsT
1)( , (D.51)
where
∏≠=
→−
=−=n
ijj
ji
ispi
issi
ss
sLsTssT
1)(
)()()(lim , i = 0,1,…,n. (D.52)
From (D.52) it follows that
1rank =iT , i = 0,1,…,n. (D.53)
We decompose the matrix Ti into the two matrices Bi and Ci of rank equal to 1
iii BCT = , 1rank rank == ii BC , i = 0,1,…,n. (D.54)
We will show that the matrices
]...[,],,...,,[ blockdiag 1
1
21 n
n
nJ CCC
B
B
BsssA =
==
(D.55)
are minimal realization of the matrix Tsp(s).
D.4 Computation of Cyclic Realizations with the Matrix A 535
To this end we compute
).(
1,...,
1,
1 blockdiag ]...[][
11
1
211
1
sTss
T
ss
BC
B
B
ssssssCCBAsIC
sp
n
i i
in
i i
ii
nn
nJn
=−
=−
=
−−−
=−
==
−
(D.56)
Thus the matrices (D.55) are a realization of the matrix Tsp(s). It is easy to check that
n
Bss
Bss
Bss
BAsI
nn
Jn =
−
−−
=−
...00
0...0
0...0
rank ][rank 22
11
(D.57)
for all C∈s since 1rank =iB for i = 0,1,…,n.
Analogously
n
CCC
ss
ss
ss
C
AsI
n
n
Jn =
−
−−
=
−
...
...00
0...0
0...0
rank rank
21
2
1
(D.58)
for all C∈s since 1rank =iC for i = 0,1,…,n.
Thus (AJ, B) is controllable pair and (AJ, C) is observable pair. Hence the realization (D.55) is minimal.
The desired cyclic realization (D.55) can be computed using the following procedure.
Procedure D.2
Step 1. Using (D.52), compute the matrices Ti for i = 1,2,…,n. Step 2. Decompose the matrices Ti into the product (D.54) of the matrices Bi
and Ci for i = 1,2,…,n. Step 3. Compute the desired cyclic realization (D.55). Example D.2. Given the normal strictly proper matrix
536 D Minimal and Cyclic Realizations and Structure Stability
+++
++=
+++
++=21
22
)2)(1(
1
1
1
)2)(1(
11
1
1
1
)(s
ss
sssss
sssTsp
(D.59)
compute its cyclic realization. In this case m(s) = (s + 1)(s + 2) and matrix (D.59) has the real poles s1 = – 1
and s2 = – 2. Using Procedure D.2 we compute.
Step 1. Using (D.52) we obtain
.01
00
1
2
1
11
2
1
2
)()(lim
,11
111
2
111
)()(lim
2
22
2
1
11
1
−
=
++
+
++
++
=−=
=
+=−=
−=
→
−=→
s
spss
s
spss
s
s
s
s
s
s
s
sTssT
ssTssT
(D.60)
Step 2. We decompose the matrices (D.60) into the products (D.54)
].01[,1
0,
01
00
],11[,1
1,
11
11
22222
11111
=
−
==
−
=
=
==
=
BCBCT
BCBCT
(D.61)
Step 3. Thus the desired cyclic realization of the matrix (D.59) is
.11
01][,
01
11,
20
01
0
021
2
1
2
1
−
==
=
=
−
−=
= CCC
B
BB
s
sAJ
(D.62)
It is easy to check that (AJ, B) (determined by (D.62) and (D.45)) is controllable pair and (AJ, C) is and observable pair. Thus the obtained realization is cyclic.
If the matrix Tsp(s) has complex conjugate poles, then using Procedure D.2 we obtain the cyclic realization (D.55) with complex entries. In order to obtain a realization with real entries we additionally transform the complex realization (D.55) by the following similarity transformation.
Let the equation m(s) = 0 have r distinct real roots s1, s2,…, sr and q distinct pair of complex conjugate roots a1 + jb1, a1 – jb1,…, aq + jbq, aq – jbq, r + q = n.
D.4 Computation of Cyclic Realizations with the Matrix A 537
Let the complex realization (D.55) have the form
.
],......[
],,,...,,,,...,,[ blockdiag
11
11
1
11111
111121
−+
−+
=
−+−+=
−+−+=
r
qqqqr
qqqqrJ
jdc
jdc
jdc
jdc
B
B
B
jhgjhgjhgjhgCCC
jbajbajbajbasssA
(D.63)
In this case the similarity transformation matrix P has the form
−
=∈= ×j
jDDDA nn
J 1
1
2
1,],...,,1,...,1[ blockdiag 111 C . (D.64)
Using (D.63) and (D.64) we obtain
==
−−==
==
−
−
q
q
r
qqr
qrJJ
d
c
d
c
B
B
BPB
hghgCCCPC
AAsssPAPA
2
2
2
2
],......[
],,...,,,...,,[ blockdiag
1
1
1
1
111
1211
(D.65)
since
].[][,2
2
,0
0
11
1
11
1
kkkkkkk
k
kk
kk
kk
kk
kk
kkk
hgDjhgjhgd
c
jdc
jdcD
ab
baD
jba
jbaDA
=−+
=
−+
−=
−
+=
−
−
(D.66)
Thus the realization (D.65) has only real entries.
538 D Minimal and Cyclic Realizations and Structure Stability
Example D.3. Given the normal matrix
+−+
+++=
24
31
243
1)( 223 ss
s
ssssTsp
(D.67)
compute its real cyclic realization (AJ, B, C). The matrix (D.67) has one real roots s1 = – 1 and the pair of complex conjugate
roots s2 = – 1 + j, s3 = – 1 – j since
243)1)(1)(1())()(( 23321 +++=++−++=−−− sssjsjssssssss . (D.68)
Applying Procedure D.2 we obtain.
Step 1. Using (D.52) we obtain
.212
11
2
1
24
31
)1)(1(
1)()(lim
,212
11
2
1
24
31
)1)(1(
1)()(lim
,21
21
24
31
)1)(1(
1)()(lim
123
33
122
22
121
11
+
+−−=
+−+
−++=−=
−−
−−−=
+−+
+++=−=
−−
=
+−+
++−+=−=
−−=→
+−=→
−=→
jj
jss
s
jsssTssT
jj
jss
s
jsssTssT
ss
s
jsjssTssT
jssp
ss
jssp
ss
ssp
ss
(D.69)
Step 2. Decomposing the matrices (D.69) into the products (D.54) we obtain
.2
11
2
1,
2
1,
212
11
2
1
,2
11
2
1,
2
1,
212
11
2
1
],21[,1
1,
21
21
33333
22222
11111
+−−=
−
==
+
+−−=
−−−=
==
−−
−−−=
=
−
==
−−
=
jBj
CBCjj
jT
jBj
CBCjj
jT
BCBCT
(D.70)
Step 3. Thus the desired cyclic realization of the matrix (D.67) with complex
entries is
D.4 Computation of Cyclic Realizations with the Matrix A 539
.221
111][
,
2
11
2
121
121
21
,
100
010
001
00
00
00
321
3
2
1
3
2
1
−−
==
+−−
−−−=
=
−−+−
−=
=
jjCCCC
j
j
B
B
B
B
j
j
s
s
s
AJ
(D.71)
In order to compute a real realization we perform the similarity transformation (D.64) on the realization (D.71)
.
2
1
2
10
2
1
2
10
001
],1[ blockdiag 1
−
==
j
jDP
(D.72)
Using (D.65) we obtain
,
110
110
001
2
1
2
10
2
1
2
10
001
100
010
001
2
1
2
10
2
1
2
10
0011
1
−−−
−=
−
−−+−
−
−
==
−
−
j
j
j
j
j
jPAPA JJ
,
10
21
21
2
11
2
12
11
2
1
21
2
1
2
10
2
1
2
10
0011
1
−−−=
+−−
−−−
−
==
−
−
j
j
j
jBPB
(D.73)
.201
011
2
1
2
10
2
1
2
10
001
221
111
−−
=
−
−−
==
j
jjj
CPC
540 D Minimal and Cyclic Realizations and Structure Stability
Let in general case
=
=−−−=p
ii
pmp
mm nmsssssssm1
22
11 ;)...()()()( , (D.74)
where si, i = 1,2,…,p are real or complex conjugate poles. In this case the matrix Tsp(s) can be expressed as
= =
+−−=
p
i
im
jjim
i
jisp
ss
TsT
1 11
,
)()( , (D.75)
where
issspim
ij
j
ji sTssds
d
jT =−
−−
−= )]()[(
)!1(
11
1
, . (D.76)
Let only one Jordan block Ji of the form (D.47) correspond to the i-th pole si with multiplicity mi and the matrices B and C have the form
]...[, 1
1
p
p
CCC
B
B
B =
= , (D.77)
where
piCCC
B
B
B imii
imi
i
i ,...,2,1],...[, ,1,
,
1,
==
= . (D.78)
Taking into account that
pi
ss
ssss
ssssss
JsI
i
imii
imiii
in ,...,2,1,
1...00
)(
1...
10
)(
1...
)(
11
][ 1
2
1 =
−
−−
−−−
=− −−
(D.79)
we obtain
imiiim
i
im
kkiki
i
im
kkiki
iiini BC
ssBC
ssBC
ssBJsIC ,1,
1
11,,2
1,,
1
)(
1...
)(
11][
−++
−+
−=−
−
=+
=
− .
(D.80)
D.4 Computation of Cyclic Realizations with the Matrix A 541
Comparison of (D.75) to (D.80) yields
=
+−=j
kkjimikiji BCT
1,,, for imjpi ,...,2,1,,...,2,1 == . (D.81)
From (D.81) for j = 1 we obtain
imiii BCT ,1,1, =
(D.82)
The matrix Ti,1 we decompose it into the column matrix Ci,1 and the row
matrix imB . Now for (D.81) with j = 2, we obtain
imiiimiii BCBCT ,2,1,1,2, += − . (D.83)
With Ti,2 and Ci,1, imB known, we take as the vector Ci,2 this column of the matrix
Ti,2 that corresponds to the first nonzero entry of the matrix imB and we multiply
it by the reciprocal of this entry. Then we compute
1,1,,2,2,)1(
2, −=−=imiiimiiii BCBCTT
(D.84)
and 1, −imiB for the known vector imiB , . From (D.81) with j = 3, we have
imiiimiiimiii BCBCBCT ,3,1,2,2,1,3, ++= −− . (D.85)
With Ti,3 and Ci,2, 1−imB known, we can compute
imiiimiiimiiii BCBCBCTT ,3,2,1,1,2,3,3, +=−= −− (D.86)
and then, in the same way as Ci,2 we can choose Ci,3 and compute 2−imB .
Pursuing further the procedure we can compute imiii CCC ,2,1, ,....,, and
imiii BBB ,2,1, ,....,, .
If the structural decomposition of the matrix L(s) of the following form is given
)()()()()( sGsmsQsPsL += (D.87)
then
pisGsssQsPsm
sLsTss im
iii
spim
i ,...,2,1),()()()()(
)()()( =−+==− ,(D.88)
542 D Minimal and Cyclic Realizations and Structure Stability
where
)(
)()(,
)(
)()(
sm
sQsQ
ss
smsm
ii
imi
i =−
= . (D.89)
Taking into account (D.89) we can write (D.76) in the form
issij
j
ji sQsPds
d
jT =−
−
−= )]()([
)!1(
11
1
, for imjpi ,...,2,1,,...,2,1 == (D.90)
since
0)]()[(1
1=− =−
−
issim
ij
jsGss
ds
d for imjpi ,...,2,1,,...,2,1 == . (D.91)
From (D.90) it follows that the matrix Ti,j depend only on the matrices P(s) and Q(s) and do not depend on the matrix G(s).
Knowing P(s) and Q(s) and using (D.90) we can compute the matrices Ti,j for i = 1,2,…,p and j = 1,2,…,mi.
It is easy to check that the matrices (AJ, B, C) determined by (D.46) and (D.77) (AJ, B) is a controllable pair and (AJ, C) is an observable pair. Thus these matrices constitute a cyclic realization. If the poles si, i = 1,2,…,sp are complex conjugate, then according to (D.64), in order to obtain a real cyclic realization one has to transform it by the similarity transformation.
From the above considerations we have the following procedure for computing the cyclic realization (AJ, B, C) for given normal strictly proper matrix Tsp(s) with multiple poles.
Procedure D.3
Step 1. Compute the poles si, i = 1,2,…,p of the matrix Tsp(s) and their multiplicities mi, i = 1,2,…,p.
Step 2. Using (D.76) or (D.90) compute the matrices Ti,j for i = 1,2,…,p and j = 1,2,…,mi.
Step 3. Using the established above procedure compute the columns
imiii CCC ,2,1, ,....,, of the matrix Ci and the row imiii BBB ,2,1, ,....,,
of the matrix Bi for i = 1,2,…,p. Step 4. Using (D.46) and (D.77) compute the desired realization (AJ, B, C).
Example D.4. Given the normal matrix
++++−+
++=
2)2)(1(
)1()1(
)2()1(
1)(
22
22 sss
ss
sssTsp
(D.92)
compute its real cyclic realization (AJ, B, C).
D.4 Computation of Cyclic Realizations with the Matrix A 543
Applying Procedure D.3 we obtain.
Step 1. The matrix (D.92) has two double real poles: s1 = − 1 , m1 = 2, s2 = − 2, m2 = 2.
Step 2. Using (D.76) we obtain
.11
00
2)2)(1(
)1()1(
)1(
1)()2(lim
,00
11
2)2)(1(
)1()1(
)1(
1)()2(lim
,11
00
2)2)(1(
)1()1(
)2(
1)()1(lim
,10
00
2)2)(1(
)1()1(
)2(
1)()1(lim
2
22
22
222
2
22
22
221
1
22
22
112
1
22
22
111
−
=
++++−+
+=
+=
−=
++++−+
+=+=
−
=
++++−+
+=
+=
=
++++−+
+=+=
−=→
−=→
−=→
−=→
s
spss
ssp
ss
s
spss
ssp
ss
sss
ss
sds
dsTs
ds
dT
sss
ss
ssTsT
sss
ss
sds
dsTs
ds
dT
sss
ss
ssTsT
(D.93) Step 3. Using (D.82) and (D.83) we obtain
.11
00
],10[,1
0,
10
00
1212111112
1211121111
BCBCT
BCBCT
+=
−
=
=
==
=
(D.94)
We choose
−
=1
012C . Thus
=−=
01
001212121111 BCTBC
and ]01[11 =B .
.11
00
],11[,0
1,
00
11
2222212122
2221222121
BCBCT
BCBCT
+=
−
=
−=
==
−=
(D.95)
We choose
−
=1
022C . Thus
=−=
00
002222222121 BCTBC
and ]00[21 =B .
544 D Minimal and Cyclic Realizations and Structure Stability
Step 4. Using (D.46) and (D.77) we obtain the desired realization
.1011
0100][
,
11
00
10
01
,
1000
1200
0010
0011
22211211
22
21
12
11
−−
==
−
=
=
−−
−−
=
CCCCC
B
B
B
B
BAJ
(D.96)
Theorem D.6. A realization ),,,( 11 DCPPBPAP −− for any arbitrary
nonsingular matrix P is a cyclic realization if and only if ),,,( DCBA is a cyclic
realization.
Proof. According to Theorem D.2 ),,,( 11 DCPPBPAP −− is a minimal
realization if and only if (A, B, C) is minimal realization. We will show that the similarity transformation does not change the invariant polynomials of A. Let U and V be the unimodular matrices of elementary operations on the rows and columns of [Ins – A] transforming this matrix to its Smith canonical form, i.e.
)(])[(][ sVAsIsUAsI nSn −=− . (D.97)
Let 1)()( −= PsUsU and )()( sPVsV = . )(sU and )(sV are also unimodular
matrices for any nonsingular matrix P since 1det)(det)(det −= PsUsU
and )(detdet)(det sVPsV = , with Pdet and 1det −P independent
of the variable s. We will show that the matrices )(sU and )(sV reduce the
matrix [Ins – PAP-1] to its Smith canonical form [Ins – A]S.
Using the definition of )(sU and )(sV , and (D.97) we obtain
Snn
nn
AsIsVAsIsU
sPVPAsIPPsUsVPAPsIsU
][)(])[(
)(][)()(])[( 111
−=−=−=− −−−
. (D.98)
Thus the matrices [Ins – PAP-1], [Ins – A] have the same invariant polynomials. Hence (PAP-1, PB, CP-1, D) is a cyclic realization if and only if (A, B, C, D) is a cyclic realization. □
D.5 Structural Stability of Cyclic Matrices and Cyclic Realizations 545
D.5 Structural Stability of Cyclic Matrices and Cyclic Realizations
D.5.1 Structural Stability of Cyclic Matrices
Definition D.4. nnA ×ℜ∈ is called a structurally stable matrix if there exists such
a positive number ε0 that for any matrix nnB ×ℜ∈ and any ε satisfying
the condition 0εε < all the matrices A + Bε are stable.
Theorem D.7. A cyclic matrix nnA ×ℜ∈ is structurally stable. The proof of this theorem it is based on the following two facts:
1) If nnA ×ℜ∈ is nonsingular matrix then all the matrices A + B are also nonsingular whenever
α<B
(D.99)
for α a certain positive number.
2) If nnA ×ℜ∈ has rank A = r, then rank [A + B] ≥ r for the matrix nnB ×ℜ∈ satisfying the condition (D.99).
Noncyclic matrices are not structurally stable but for a noncyclic matrix nnA ×ℜ∈ one can always choose a matrix nnB ×ℜ∈ and a sufficiently small
number )0( >εε so that the sum A + Bε is a cyclic matrix.
Only for a particular choice of the matrix B and ε the sum A + Bε is a noncyclic matrix. The matrix in Frobenius canonical form
−−−−
=
−1210 ...
1...000
0...100
0...010
naaaa
A
(D.100)
is a cyclic matrix regardless of the values of the coefficients ai, i = 0,1,…,n – 1. For example the matrix
=
a
A
00
010
011
(D.101)
is a cyclic matrix for all the values of the coefficient a ≠ 1, and it is a noncyclic matrix only if a = 1.
546 D Minimal and Cyclic Realizations and Structure Stability
Let nnA ×ℜ∈Δ be regarded as a disturbance (uncertainty) to the nominal
matrix nnA ×ℜ∈ , and take εB = ΔA. Then, according to Theorem D.7, since A is cyclic, the matrix A + ΔA is also cyclic.
D.5.2 Structural Stability of Cyclic Realization
A minimal realization ),,,( DCBA with the matrix A cyclic is called a cyclic
realization.
Theorem D.8. Let ),,,( 1111 DCBA be a cyclic realization and ),,,( 2222 DCBA
be another realization of the same dimensions. Then there exist such a number ε0 > 0 that all the realizations
pmnRDDCCBBAA ,,21212121 ),,,( ∈++++ εεεε for 0εε <
(D.102)
are cyclic realizations.
Proof. According to Theorem D.7, if A1 is a cyclic matrix, then all the matrices
A1 + εA2 are cyclic for 0εε < . If (A1, B1) is a controllable pair then (A1 + εA2,
B1 + εB2) is also controllable for all 1εε < . Analogously, if (A1, C1)
is an observable pair then (A1 + εA2, C1 + εC2) is also observable for all 2εε < .
Thus the realization (A1 + εA2, B1 + εB2, C1 + εC2) is a minimal one for
021 ),min( εεεε =< , and with (A1 + εA2) being cyclic matrix it is a cyclic
realization as well. □
Example D.5. A cyclic realization ),,( 111 CBA is given with
]001[,
1
0
0
,100
010
11
121110
1 =
=
= CB
aaa
A , (D.103)
where a10, a11, a12 are arbitrary parameters. For which value of the parameters a20, a21, a22, b and c in the matrices
]00[,0
0
,100
010
22
222120
2 cC
b
B
aaa
A =
=
= (D.104)
the realization ),,( 212121 CCBBAA +++ is a cyclic one?
D.5 Structural Stability of Cyclic Matrices and Cyclic Realizations 547
We denote
]01[,
1
0
0
,200
020
2121
210
21 cCCC
b
BBB
aaa
AAA =+=
+=+=
=+= ,
(D.105) where
kkk aaa ,2,1 += for k = 0,1,2. (D.106)
A is a cyclic matrix for all the values of the parameters a20, a21 and a22. (A, B) is a controllable pair for those values of the parameters a20, a21, a22 and b, for which det [B AB A2B] ≠ 0, that is
0)1(8
)1)(2()1()1(
)1(2)1(20
)1(400
]det[ 3
2212
22 ≠+=
++++++
+= b
baabab
bab
b
BAABB for b ≠ – 1
(D.107)
(A, C) is a observable pair for those values of the parameters a20, a21, a22 and c, for which
0det2
≠
CA
CA
C
(D.108)
that is
0)2(4
2422
220
01
det 21
302
2102
≠−++=+
=
cacaca
cacac
c
c
CA
CA
C
(D.109)
for 302
21 2 cacaca ++≠ and taking (D.106) into account we obtain
2123
102
11222
213
20 −−+≠+− cacacacacaca . (D.110)
Thus (A, B, C) is a cyclic realization for the parameters a20, a21, a22, b and c in the matrices (D.104) satisfying the condition (D.110) and b ≠ – 1.
548 D Minimal and Cyclic Realizations and Structure Stability
D.5.3 The Impact of the Coefficients of the Transfer Function on System Description
Consider the transfer matrix
++
+++
=as
s
sssT
10
02
)2)(1(
1)( . (D.111)
This matrix is normal if and only if a = 0, since the polynomial
)2)(1(10
02+++=
+++
sasas
s
(D.112)
is divisible without reminder by (s + 1)(s + 2) if and only if a = 0. For a = 0 there exist a cyclic realization ),,( CBA of the matrix (D.111) with
=
=
−
−=
10
01,
10
01,
20
01CBA (D.113)
which can be computed using Procedure D.2. Applying Procedure D.2 for a ≠ 0 we obtain
].10[,1
0
,10
00
10
02
1
1)()(lim
,0
01,
10
01
,0
01
10
02
2
1)()(lim
22
222
22
2
11
111
11
1
aBC
BCaas
s
ssTssT
aBC
BCaas
s
ssTssT
sss
sss
−=
=
=
−
=
++
++
=−=
=
=
=
=
++
++
=−=
−=→
−=→
(D.114)
Thus the desired minimal realization is
==
−=
=
−−
−=
=
110
001][,
10
0
01
,
200
010
001
0
021
2
1
2
12 CCC
a
aB
BB
s
sIA .
(D.115)
To the cyclic realization (D.113) corresponds a system described by the following state equations
D.5 Structural Stability of Cyclic Matrices and Cyclic Realizations 549
.
,
,2
,
22
11
222
111
xy
xy
uxx
uxx
==
+−=+−=
(D.116)
To the minimal realization (D.115) corresponds a system described by the following state equations
.
,
,)1(2
,
,
322
11
233
222
111
xxy
xy
uaxx
auxx
uxx
+==
−+−=+−=
+−=
(D.117)
Note that for a = 0 in (D.117) we do not obtain (D.116), and the pair (A, B) of the system (D.117) becomes not controllable.
The above considerations can be generalized into the case of linear systems of any order.
The computation of normal transfer matrix on the basis of its approximation is addressed in [100].
Consider the transfer matrix
)()(
)()( s
sm
sLsT mp
p×ℜ∈=
(D.118)
whose coefficients differ from the coefficients of a normal transfer matrix
)()(
)()( s
sm
sLsT mp×ℜ∈= . (D.119)
The problem of computing the normal transfer function on the basis of its approximation can be formulated in the following way.
With the transfer matrix (D.118) given, one has to compute the normal transfer matrix (D.119), which is a good approximation of the matrix (D.118).
Below we provide a method of solving the problem. The method is based on the structural decomposition of the matrix (D.118).
Applying elementary operations we transform the polynomial matrix
][)( ssL mp×ℜ∈ into the form
=
)()(
)(1)()()()(
sMsk
swsisVsLsU , (D.120)
where U(s) and V(s) are polynomial matrices of elementary operations on rows and column, respectively, i(s) is a polynomial and
550 D Minimal and Cyclic Realizations and Structure Stability
][)(],[)(],[)( )1()1(1)1(1 ssMsskssw mppm −×−−−× ℜ∈ℜ∈ℜ∈ . (D.121)
Premultiplication of the matrix
=
)()(
)(1)(1 sMsk
swsL
(D.122)
by the unimodular matrix
−
=−
−
1
1,11 )(
01)(
p
p
IsksU
(D.123)
and postmultiplication by the unimodular matrix
−=
−− 11,11 0
)(1)(
mm I
swsV
(D.124)
yields
−
=−
−)()()(0
01)()()(
1,1
1,1111 swsksM
sVsLsUp
m. (D.125)
In the method we take
)()()()()()( 1 sRsMsmswsksM +=− , (D.126)
where
)(deg)(deg],[)(],[)( )1()1()1()1(1 smsRssRssM mpmp <ℜ∈ℜ∈ −×−−×− .
(D.127)
In the further considerations we omit the polynomial matrix R(s). From (D.120) and (D.125) we have
).(0
)(1
)()()(0
01
)(
01)()(
)()()()()(0
01)()()(
)()()()()(
1
11,11,1
1,1
1
1,11
111
1,1
1,111
1
11
1
sVI
sw
swsksMIsksisU
sVsVswsksM
sUsisU
sVsLsisUsL
mmp
m
p
p
p
m
−
−−−
−
−
−−
−−
−
−−−
−−
−
=
−
=
=
(D.128)
D.5 Structural Stability of Cyclic Matrices and Cyclic Realizations 551
Using (D.126), (D.128) and omitting R(s) we obtain
)(0
)(1
)()(0
01
)(
01)()()( 1
11,111,1
1,1
1
1,11 sVI
sw
sMsmIsksisUsL
mmp
m
p
p −
−−−
−
−
−−
=
.(D.129) and
)()(
)()(
)(
)()( sG
sm
sQsP
sm
sLsT +== , (D.130)
where
).()(0
00)()()(
),(])(1[)(,)(
1)()()(
1
11,1
1,11
11
sVsM
sUsisG
sVswsQsk
sUsisP
p
m −
−
−−
−−
=
=
=
(D.131)
The above considerations yields the following procedure for solving our problem.
Procedure D.4 Step 1. Applying elementary operations transform the matrix )(sL into the form
(D.120) and compute the polynomial i(s) as well as the unimodular matrices U(s) and V(s).
Step 2. Choose M1(s) and R(s). Step 3. Using (D.131) compute the matrices P(s), Q(s) and G(s). Step 4. Using (D.130) compute the desired normal transfer matrix T(s).
Example D.6. Provided the parameter a is small enough (close to zero), compute the normal transfer matrix for the matrix (D.111). Step 1. In this case )2)(1()()( ++== sssmsm and
++
+=
as
ssL
10
02)( . (D.132)
Applying the elementary operations L[1 + 2] and )]1(21[ −×+P we
obtain
.
1
1
1
)1(1
11
)1(1)1(
11
11
01
10
02
10
11)()()(
−++
−++−
−++
−=
++++−++−
=
−
++
+
=
a
as
a
asa
as
aasas
asa
as
ssVsLsU
(D.133)
552 D Minimal and Cyclic Realizations and Structure Stability
Thus
.1
1)(,
1
1)(,
1
1)(
,11
01)(,
10
11)(
,11
01)(,
10
11)(),1()(
11
a
assM
a
assk
a
assw
sVsU
sVsUasi
−++=
−++−=
−++=
=
−=
−
=
=−=
−−
(D.134)
Step 2. In this case
).()()()1(
)2()2)(1(
)1(
)1(
1
1)()()(
12
2
2
sRsMsma
sass
a
as
a
asswsksM
+=−
++++=
−+++
−++=−
(D.135)
We take
21)1(
1)(
asM
−= and
2)1(
2)(
a
sasR
−+= . (D.136)
Step 3. Using (D.131) we obtain
.11
11
1
111
01
)1(
10
00
10
11)1(
)()(0
00)()()(
,1
1
1
211
01
1
11)(])(1[)(
,)1(
2
11
1
10
11)1(
)(
1)()()(
2
1
1
1
1
1
−−−
=
−
−−=
=
−++
−+=
−++==
++−
+=
−++−
−−=
=
−−
−
−
aaa
sVsM
sUsisG
a
as
a
s
a
assVswsQ
as
s
a
asask
sUsisP
(D.137)
Step 4. Thus the desired matrix is
−−−+−
−
−−+
++=+=
a
aaas
a
aa
a
a
s
sssG
sm
sQsPsT
1
21)21(
1
11
2
)2)(1(
1)(
)(
)()()( 2 .
(D.138)
D.5 Structural Stability of Cyclic Matrices and Cyclic Realizations 553
Note that for a = 0 in (D.138) we obtain the normal transfer matrix
+
+++
=+=10
02
)2)(1(
1)(
)(
)()()(
s
s
sssG
sm
sQsPsT
(D.139)
which can be also obtained form (D.111) for a = 0.
Appendix E Positive 2D Continuous-Discrete Linear Systems
E.1 General Model of Continuous-Discrete Linear Systems and Its Solution
E.1 General Model of Cont inuous-Discrete Linear Systems and Its So lut ion
Following [105] let us consider the general model of linear continuous-discrete system described by the equations
,...},1,0{],,0[),,(),(),(
),1,(),(),()1,(),(),()1,( 210210
=∈+∞=ℜ∈+=+++++++=+
++ ZititDuitCxity
ituBituBituBitxAitxAitxAitx
(E.1)
where t
itxitx
∂∂= ),(
),( , nitx ℜ∈),( , mitu ℜ∈),( , pity ℜ∈),( are the state,
input and output vectors and ,nnkA ×ℜ∈ ,mn
kB ×ℜ∈ ;2,1,0=k ,npC ×ℜ∈
mpD ×ℜ∈ . Boundary conditions for (E.1) are given by
),0(0 ixx i = , +∈ Zi and )0,(0 txxt = , )0,(0 txxt = , +ℜ∈t . (E.2)
The transition matrix of the model (E.1) is defined as follows
0or0
,;0
0
for
for
for
02,111,01,1,121,11,10
<<∈>+
==
++=++= +−−−−−−−−
ji
Zjiji
ji
ATATATTATATA
I
T jijijijijiji
n
ij
(E.3)
556 E Positive 2D Continuous-Discrete Linear Systems
Theorem E.1. The solution of the equation (E.1) with boundary conditions (E.2) has the form
).0,()0,()!1(
)(
)0,0(!
)0,(!
)()0,(
!
)(
),(),()!1(
)(
),0(!
),0(!
),0(!
),(!
)(),(
!
)(
),(
,01 0
1
,
0,
02,
02,
011,0
1 0 0
1
11,
0 011,,11,
02,
001,
txTdxk
tT
xk
tT
dxk
tATdu
k
tBT
ltuBTdluk
tBT
lxk
tATlx
k
tTlu
k
tBT
dluk
tBTdlu
k
tBT
itx
ik
t k
ik
k k
ik
t k
ik
t k
ik
lli
k l
t k
lik
k l k
lik
k
lik
k
lik
t k
lik
t k
lik
+
−−+
+
−+−
−
+
−−+
−+−
−+−
=
∞
=
−
∞
=
∞
=−−
∞
=
∞
=
−−−
∞
=
∞
=−−−−−
−−−
τττ
ττττττ
τττ
ττττττ
(E.4)
Proof is given in [105]. Knowing the matrices Ak, Bk, k = 0,1,2 of (E.1), boundary conditions (E.2) and
input u(t,i), +ℜ∈t , +∈ Zi we can compute the transition matrices (E.3) and
using (E.4) the state vector x(t,i) for +ℜ∈t , +∈ Zi . Substituting the state vector
into (E.1) we can find the output vector y(t,i) for +ℜ∈t , +∈ Zi .
E.2 Positive General Model of Continuous-Discrete Linear Systems
Definition E.1. The general model (E.1) is called positive if nitx +ℜ∈),(
and pity +ℜ∈),( , +ℜ∈t , +∈ Zi for any boundary conditions
nt
nt
ni xxx +++ ℜ∈ℜ∈ℜ∈ 000 ,, pity +ℜ∈),( , +ℜ∈t and all inputs
mitu +ℜ∈),( , mitu +ℜ∈),( , +ℜ∈t , +∈ Zi .
Theorem E.2. The general model (E.1) is positive if and only if
E.2 Positive General Model of Continuous-Discrete Linear Systems 557
.,,,,
,,,,
210
210102mpnpmn
nnnnn
DCBBB
AAAAAMA
×+
×+
×+
×+
×+
ℜ∈ℜ∈ℜ∈
ℜ∈+ℜ∈∈ (E.5)
Proof is given in [105]. Consider the general 2D model
1,2,11,01,2,11,01,1 ++++++ +++++= jijijijijijiji uBuBuBxAxAxAx , (E.6)
where njix ℜ∈, , m
jiu ℜ∈, are the state and input vectors and
,nnkA ×ℜ∈ ,mn
kB ×ℜ∈ +∈ Zji, , 2,1,0=k .
Definition E.2. The model (E.6) is called positive if njix +ℜ∈, , +∈ Zji, for all
boundary conditions
n
ix +ℜ∈0 , +∈ Zi and njx +ℜ∈0 , +∈ Zj (E.7)
and every input mjiu +ℜ∈, , +∈ Zji, .
Theorem E.3. The model (E.6) is positive if and only if
mn
knn
k BA ×+
×+ ℜ∈ℜ∈ , for k = 0,1,2. (E.8)
It is well-known that the transition matrix Tij (defined also by (E.3)) of the positive
model (E.6) is a positive matrix, i.e. nnijT ×
+ℜ∈ for +∈ Zji, . Note that the
transition matrix Tij of the positive model (E.1) may be not always a positive matrix. For example for the model (E.1) with the matrices
2,1,0,,21
21,
12
01,
01
12 22210 =ℜ∈
−
−=
=
= ×
+ kBAAA k , (E.9)
we have
.20
31 22210
×+ℜ∈
=+= AAAA
(E.10)
Therefore, by Theorem E.2 the model with the matrices (E.9) is positive, but the matrices
−
−==
−
=++=63
63,
03
54 22201221011 ATAAAAAT
(E.11)
have some negative entries.
558 E Positive 2D Continuous-Discrete Linear Systems
Remark E.1. From (E.6) it follows that if A2 = 0 then the general model (E.1) of the continuous-discrete systems is positive if and only if the general 2D model (E.6) is positive.
E.3 Reachability of the Standard and Positive General Model
Definition E.3. The model (E.1) is called reachable at the point (tf, q) if for any
given final state nfx ℜ∈ there exists an input u(t, i), qitt f ≤≤≤≤ 0,0
which steers the system form zero boundary conditions to the state xf, i.e. x(tf, q) = xf.
Theorem E.4. The model (E.1) is reachable at the point (tf, q) for tf > 0 and q = 1 if and only if one of the following conditions is satisfied.
1) C∈∀=−⇔=− snBAsInBABAB nn ][rank]...[rank 020
12020 ,
2) the rows of the matrix 02 Be tA are linearly independent over the field
of complex numbers C.
Proof is given in [105].
Theorem E.5. The model (E.1) is reachable at the point (tf, q) for tf > 0 and q = 1 if and only if the matrix
0,0
200
2 >= f
ft TATAf tdeBBeR τττ
(E.12)
is positive definite (nonsingular). Moreover, the input which steers the system from zero boundary conditions to xf is given by
.)0,( 1)(2
0 fftftTAT xReBtu −−
=
(E.13)
Proof is given in [105].
Remark E.2. Reachability is independent of the matrices A0, A1, B1, B2.
Remark E.3. To simplify the calculation we may assume that u(t,0) is piecewise constant (is the step function).
Example E.1. Consider the general model (E.1) with the matrices
=
=
0
1,
21
0102 BA
(E.14)
E.3 Reachability of the Standard and Positive General Model 559
and arbitrary remaining matrices of the system. Applying the condition 2) of Theorem E.4 we obtain
2
10
11][rank 020 =
=BAB
(E.15)
and
C∈∀=
−−
−=− s
s
sBAsI 2
021
101rank][rank 022 . (E.16)
Therefore, the system (E.1) with matrices (E.14) is reachable for q = 1 and tf > 0. Assuming tf = 2 and
=
1
2fx
(E.17)
from (E.13) and (E.12) we may find the input that steers the system for zero boundary conditions to the desired state (E.17)
.0953.05519.0)0,( 2421)(2
0tt
ffftTAT eexReBtu −−−−
−==τ
(E.18)
The plots of the state variables for q = 1, ]2,0[∈t and input for q = 0 and
]2,0[∈t are shown in Figure E.1 and Figure E.2, respectively.
Fig. E.1 State variables of the system with )0,(tu of the form (E.18)
560 E Positive 2D Continuous-Discrete Linear Systems
Fig. E.2 Input (E.18) of the system
Let us assume, that the input of the system is piecewise constant, i.e.
≤≤<≤
=fttt
tt
u
utu
1
1
2
1 0
for
for)0,(
(E.19)
where u1 and u2 are constant values. Taking into account (E.14) and (E.17) we obtain
.1
1
1
2
10
11][
)(
)( 11
0201
0
=
==
−−
ff
fxBAB
tv
tv
(E.20)
For (E.19) we have
.1
1
)()(
)()(
1
1
1
1
01
1
0
1
00
2
1
−−
−−
=
−
ft
tf
t
f
ft
tf
t
f
dtcdtc
dtcdtc
u
u
ττττ
ττττ
(E.21)
Using −
==
1
02
2 )(n
kfk
kftAtcAe it is easy to show that
tt eetc 2
0 2)( −= , tt eetc −= 21 )( . (E.22)
E.3 Reachability of the Standard and Positive General Model 561
Using formula (E.21) we may compute values of the system input for arbitrary t1 and tf ( ftt << 10 ). For t1 = 1 and tf = 2, we obtain
.2948.1
0481.0
2
1
−=
u
u
(E.23)
The plots of state variables and input for q = 1 and ]2,0[∈t are shown in Figure E.3 and Figure E.4, respectively.
Fig. E.3 State variables of the system with )0,(tu of the form (E.19)
Fig. E.4 Input (E.19) of the system
562 E Positive 2D Continuous-Discrete Linear Systems
Definition E.4. The positive system (E.1) is called reachable at the point (tf , q) if
for any given final state nfx +ℜ∈ there exists a nonnegative input mitu +ℜ∈),( ,
qitt f ≤≤≤≤ 0,0 which steers the system from zero boundary conditions to
the state xf , i.e. x(tf , q) = xf.
Theorem E.6. The positive model (E.1) is reachable at the point (tf , q) for tf > 0 and q = 1 if the matrix
0,0
200
2 >= f
ft TATAf tdeBBeR τττ
(E.24)
is a monomial matrix. The input that steers the system in time tf from zero boundary conditions to the state xf is given by the formula
.)0,( 1)(20 ff
tftTAT xReBtu −−=
Proof is given in [105].
E.4 Stability of the Positive General Model
Consider the 2D continuous-discrete linear system (first Fornaisni-Marchesini model)
),()1,(),(),()1,( 210 itBuitxAitxAitxAitx ++++=+ , +ℜ∈t , +∈ Zi ,
(E.25)
where t
itxitx
∂∂= ),(
),( , nitx ℜ∈),( , mitu ℜ∈),( nnAAA ×ℜ∈210 ,, ,
mnB ×ℜ∈ .
Definition E.5. The positive model (E.25) is called asymptotically stable if for 0),( =itu
0),(lim
,=
∞→itx
it (E.26)
for any bounded boundary conditions.
The matrix nnA ×ℜ∈ is called asymptotically stable (Hurwitz) if all its eigenvalues lie in the open left half of the complex plane.
Definition E.6. The point xe is called equilibrium point of the asymptotically
stable system (E.25) if for nTnBu +ℜ∈== ]1...1[1
E.4 Stability of the Positive General Model 563
nee xAxA 10 20 ++= . (E.27)
Asymptotic stability implies 0]det[ 20 ≠+ AA and from (E.27) we have
ne AAx 1][ 120
−+−= . (E.28)
Remark E.4. From (E.25) for B = 0 it follows that the positive system is asymptotically stable only if the matrix nIA −1 is Hurwitz Metzler matrix.
In what follows it is assumed that the matrix nIA −1 is a Hurwitz Metzler
matrix.
Theorem E.7. The linear continuous-discrete positive 2D system (E.25) is asymptotically stable if and only if all coefficients of the polynomial
000110
1,1
11,
210
...
)]1()1(det[
azasazsazsazs
zAsAAzsI
nnnn
nnnn
nn
n
++++++=
+−−−+−
−−
− (E.29)
are positive, i.e.
0, >lka for k, l = 0,1,…,n ( 1, =nna ). (E.30)
Proof is given in [105].
Theorem E.8. Let the matrix nIA −1 be a Hurwitz Metzler matrix. The positive
continuous-discrete linear 2D system (E.25) is asymptotically stable if and only if
there exists a strictly positive vector n+ℜ∈λ (all components of the vectors are
positive) such that
0)( 20 <+ λAA . (E.31)
Proof is given in [105].
Remark E.5. As the strictly positive vector λ we may choose the equilibrium point (E.28) since for ex=λ we have
nnAAAAAA 11))(()( 1202020 −=++−=+ −λ . (E.32)
Theorem E.9. The positive system (E.25) is asymptotically stable if and only if both matrices
nIA −1 , 20 AA + (E.33)
are Hurwitz Metzler matrices.
564 E Positive 2D Continuous-Discrete Linear Systems
Proof. From Remark E.4 it follows that the positive system (E.25) is asymptotically stable only if the matrix nIA −1 is Hurwitz Metzler matrix. By
Theorem E.8 the positive system is asymptotically stable if and only if there exists a strictly positive vector λ such that (E.31) holds but this is equivalent that the matrix 20 AA + is Hurwitz Metzler matrix. □
To test of the matrices (E.33) are Hurwitz Metzler matrices the following theorem is recommended.
Theorem E.10. The matrix nnA ×ℜ∈ is a Hurwitz Metzler matrix if and only if one of the following equivalent conditions is satisfied:
1) all coefficients 10 ,..., −naa of the characteristic polynomial
011
1 ...]det[ asasasAsI nn
nn ++++=− −
− (E.34)
are positive, i.e. 0≥ia , i = 0,1,…,n – 1,
2) the diagonal entries of the matrices
)(kknA − for k = 1,…,n – 1 (E.35)
are negative, where
]...[,
,
...
...
...
],...[,
,
...
...
...
,
...
...
...
)(1,
)(1,
)(1
)(,1
)(,1
)(1
)(,
)(1
)(1
)(1
)(,
)(1,
)(,1
)(11
)1(1,1
)1()1()1()(
)0(1,
)0(1,
)0(1
)0(,1
)0(,1
)0(1
)0(1,1
)0(1,1
)0(1,1
)0(11
)0(1)0(
,)0(1
)0(1
)0(1
)0(,
)0(1,
)0(,1
)0(11
)0(
kknkn
kkn
kkn
kknkn
kkn
kkn
kknkn
kkn
kkn
kkn
kknkn
kkn
kkn
k
kknkn
kkn
kknn
knk
kn
nnnn
nn
n
n
nnn
n
nnnn
nn
nnn
n
n
aac
a
a
b
ac
bA
aa
aa
a
cbAA
aac
a
a
b
aa
aa
Aac
bA
aa
aa
AA
−−−−−−
−−−
−
−−
−−−−
−−−−
−−−
−
−+−+−
−−
−−−
−−
−−
−
−
−−−
−
−−
−−
=
=
=
=−=
=
=
=
=
==
(E.36)
for k = 0,1,…,n – 1.
E.4 Stability of the Positive General Model 565
To check the stability of the positive system (E.25) the following procedure can be used.
Procedure E.1
Step 1. Check if at least one diagonal entry of the matrix nnA ×+ℜ∈1 is equal
or greater then 1. If this holds then positive system (E.25) is unstable. Step 2. Using Theorem E.10 check if the matrix nIA −1 is Hurwitz Metzler
matrix. If not the positive system (E.25) is unstable. Step 3. Using Theorem E.10 check if the matrix 20 AA + is Hurwitz Metzler
matrix. If yes the positive system (E.25) is asymptotically stable.
Example E.2. Consider the positive system (E.25) with the matrices
−
−=
=
=
6.02.0
1.05.0,
3.01.0
2.04.0,
3.01.0
1.02.0210 AAA . (E.37)
By Theorem E.2 the system is positive since nMA ∈2 , nnAA ×+ℜ∈10, and
22210 13.011.0
02.004.0 ×+ℜ∈
=+ AAA .
Using Procedure E.1 we obtain the following
Step 1. All diagonal entries of the matrix A1 are less then 1. Step 2. The matrix nIA −1 is Hurwitz since the coefficient of the polynomial
4.03.17.01.0
2.06.0]det[ 2
12 ++=+−
−+=+− ss
s
sIAsI n
(E.38)
are positive. Step 3. The matrix
−
−=+=
3.03.0
2.03.020 AAA
(E.39)
is also Hurwitz since (using condition 2) of Theorem E.10)
01.0
3.03.0*2.0
3.0)1(1 <−=+−=A . (E.40)
By Theorem E.9 the positive system (E.25) with (E.37) is asymptotically stable.
566 E Positive 2D Continuous-Discrete Linear Systems
The polynomial (E.29) for positive system has the form
.03.031.04.026.028.026.11.13.1
)1(6.03.03.0)1()1(2.01.01.0
)1(1.02.01.0)1(5.04.02.0)1(
)]1()1(det[
222222
2102
++++++++=
++−−++−−−+−−−++−−+
=
+−−−+
sszzszszzszs
zszszs
zszszs
zAsAAzsI
(E.41)
All coefficients of the polynomial are positive. Therefore, by Theorem E.9 the positive system is also asymptotically stable.
It is well-known that substituting 00 =A , 0=B in (E.25) we obtain the
autonomous second Fornasini-Marchesini continuous-discrete linear 2D system
)1,(),()1,( 21 ++=+ itxAitxAitx , +ℜ∈t , +∈ Zi . (E.42)
The autonomous Roesser type continuous-discrete model has the form
=
+ ),(
),(
)1,(
),(
2221
1211
itx
itxAA
AA
itx
itxv
h
v
h, +ℜ∈t , +∈ Zi , (E.43)
where t
itxitx
∂∂= ),(
),( , 1),( nh itx ℜ∈ and 2),( nv itx ℜ∈ are the horizontal
and vertical vectors and lnknklA ×ℜ∈ , 2,1, =lk . The model (E.43) is positive
if and only if 11A is a Metzler matrix and 1221
2112 , nnnn AA ×
+×
+ ℜ∈ℜ∈ ,
2222
nnA ×+ℜ∈ . The positive model (E.43) is a particular case of the model (E.42)
for
=
=
00,
00 12112
22211
AAA
AAA . (E.44)
Theorem E.11. The positive Roesser type continuous-discrete model (E.43) is asymptotically stable if and only if the coefficients of the polynomial
00011011211
2,11121
12,121
22221
12111
ˆˆˆˆ...ˆˆ
)1(
)1()1()1(det
azasaszazsazsazs
sAzsIsA
zAzAzsI
nnnn
nnnn
nn
n
n
+++++++=
−+−+−+−+
−−
−−
(E.45)
are positive. Proof is given in [105].
E.4 Stability of the Positive General Model 567
Theorem E.12. The positive continuous-discrete 2D linear system (E.25) is unstable if one of the following conditions is satisfied
1) 0)](det[ 20 ≤+− AA ,
2) 0]det[ 2 ≤−A
3) 0]det[ 1 ≤− AIn .
Proof is given in [105].
Example E.3. Consider the positive scalar model (E.43) with
=
=
00,
00 12112
22211
aaA
aaA , 0,0,0,0 22211211 ≥≥≥< aaaa .
(E.46)
The polynomial (E.45) for (E.46) has the form
saaaaaszaaaaa
saszazsazssazssa
zazazs
)()2(
)1()2()1(
)1()1()1(det
11211222112112221111
222
211
222
22
2221
1211
−−+−+−+
−+−−+=
−+−
+−+−+
(E.47)
and its coefficients are positive if and only if 10,0 2211 <≤< aa and
1121122211 aaaaa >− .
Theorem E.13. The positive linear continuous-discrete 2D system (E.25) is asymptotically stable if and only if all coefficients of the polynomial
000110
1,1
11,
210
...
)]1()1(det[
azasazsazsazs
zAsAAzsI
nnnn
nnnn
nn
n
++++++=
+−−−+−
−−
− (E.48)
are positive, i.e.
0, >lka for k, l = 0,1,…,n ( 1, =nna ). (E.49)
Proof is given in [105].
Example E.4. Consider the positive system (E.25) with the matrices
−
−=
=
=
4.02.0
1.03.0,
3.01.0
1.02.0,
4.04.0
3.05.0210 AAA . (E.50)
The matrices (E.50) satisfy the conditions (E.5) since
568 E Positive 2D Continuous-Discrete Linear Systems
22210 29.043.0
28.046.0 ×+ℜ∈
=+ AAA
(E.51)
and then the system is positive. Using (E.50) we obtain
24.006.0
4.02.0det)](det[ 20 −=
−
−−=+− AA ,
1.04.02.0
1.03.0det]det[ 2 =
−
−=− A ,
55.0
7.01.0
1.08.0det]det[ 1 =
−
−=− AIn
(E.52)
and the condition 1) of Theorem E.12 is satisfied. Therefore, the positive system (E.25) with (E.50) is unstable.
In this case the polynomial (E.48) has the form
24.022.024.03.01.055.07.05.1
4.07.06.02.01.0
4.01.01.02.03.08.0det)]1()1(det[
222222
210
−−−++−++=
++−−−
−−−−++=+−−−+
zsszzsszzszs
zsszzs
zszsszzAsAAzsIn
(E.53)
and by Theorem E.13 the system is also unstable. The robust stability of scalar linear continuous-discrete linear systems has been
investigated by M. Busłowicz in [10].
E.5 Continuous-Discrete Linear Systems with Delays
Following [55] let us consider a continuous-discrete system with q delays in state vector described by the equations
,],,0[),,(),(),(
),1,(),(),(
)1,(),(),()1,(
210
02
01
00
++
===
∈+∞=ℜ∈+=++++
+−+−+−−=+
ZititDuitCxity
ituBituBituB
ikdtxAkitxAkikdtxAitxq
k
kq
k
kq
k
k
(E.54)
where t
itxitx
∂∂= ),(
),( , nitx ℜ∈),( , mitu ℜ∈),( , pity ℜ∈),( and
,nnklA ×ℜ∈ ;2,1,0=l ;,...,1,0 qk = ,mnB ×ℜ∈ ,npC ×ℜ∈ mpD ×ℜ∈ are
the real matrices, 0>d is a delay.
E.5 Continuous-Discrete Linear Systems with Delays 569
Boundary conditions for (E.54) have the form
),(0 itx i , ]0,[ qdt −∈ , +∈ Zi , ),(0 itxt , ),(0 itxt , ]0,[ qi −∈ , +ℜ∈t (E.55)
and
0)0,(0 =tx i , ]0,[ qdt −∈ , 0),0(),0( 00 == ixix tt , ]0,[ qi −∈ . (E.56)
Definition E.7. The continuous-discrete linear system with delays (E.54) is called
(internally) positive if nitx +ℜ∈),( and pity +ℜ∈),( , +ℜ∈t , +∈ Zi for
arbitrary boundary conditions ni itx +ℜ∈),(0 , ]0,[ qdt −∈ , +∈ Zi and
nt itx +ℜ∈),(0 , n
t itx +ℜ∈),(0 , ]0,[ qi −∈ , +ℜ∈t and all inputs mitu +ℜ∈),( ,
+ℜ∈t , +∈ Zi .
Theorem E.14. The continuous-discrete linear system with delays (E.54) is internally positive if and only if
.,,,
;,...,2,1,,;,...,1,0,,,,
02
01
00
21002
mpnpmnnn
nnknnkkn
DCBAAA
qkAqkAAMA
×+
×+
×+
×+
×+
×+
ℜ∈ℜ∈ℜ∈ℜ∈+
=ℜ∈=ℜ∈∈
(E.57)
Proof is given in [55].
Definition E.8. The continuous-discrete linear system with delays (E.54) is called asymptotically stable if
0),(lim
,=
∞→itx
it (E.58)
for any bounded boundary conditions and for 0),( =itu , 0≥t +∈ Zi .
The matrix nnA ×ℜ∈ is called asymptotically stable (Hurwitz) if all its eigenvalues lie in the open left half of the complex plane.
Definition E.9. The point xe is called equilibrium point of the asymptotically
stable system (E.54) for nTnBu +ℜ∈== ]1...1[1 if the equation
nee xAxA 10 20 ++= (E.59)
is satisfied, where =
=q
k
kAA0
00 , =
=q
k
kAA0
22 . Asymptotic stability implies
0]det[ 20 ≠+ AA and from (E.59) we have
ne AAx 1][ 120
−+−= . (E.60)
570 E Positive 2D Continuous-Discrete Linear Systems
Theorem E.15. The linear continuous-discrete time system (E.54) for q = 0 is asymptotically stable if and only if the zeros of the polynomial
000110
1,1
11,
210
...
]det[
azasazsazsazs
zAsAAszI
nnnn
nnnn
nn
n
++++++=
−−−−
−−
− (E.61)
are located in the left half of the complex plane s and in the unit circle of the complex plane z.
Theorem E.16. The positive linear system
Axx = , nMA∈ (E.62)
is asymptotically stable if and only if the characteristic polynomial
011
1 ...]det[ asasasAsI nn
nn ++++=− −
− (E.63)
has positive coefficients, i.e. 0>ka for k = 0,1,…,n – 1.
Lemma E.1. A nonnegative matrix nnA ×+ℜ∈ is asymptotically stable
(nonnegative Shur matrix) if and only if the Metzler matrix nIA − is
asymptotically stable (Metzler Hurwitz matrix).
Theorem E.17. The linear continuous-discrete positive system with delays (E.54)
is asymptotically stable if and only if all coefficients of the polynomial
0001101
,11
1,
210
...
)]1()1(det[
azasazsazsazs
zAsAAzsI
nnnn
nnnn
nn
n
++++++=
+−−−+−
−−
−
(E.64)
are positive, i.e. 0, >lka for k, l = 0,1,…,n ( 1, =nna ).
Proof is given in [55].
Theorem E.18. Let the matrix nIA −1 be a Hurwitz Metzler matrix. The positive
continuous-discrete linear system with delays (E.54) is asymptotically stable if
and only if there exists a strictly positive vector n+ℜ∈λ (all components of the
vectors are positive) such that
0)( 20 <+ λAA , (E.65)
where =
=q
k
kAA0
00 , =
=q
k
kAA0
11 , =
=q
k
kAA0
22 .
Proof is given in [55].
E.5 Continuous-Discrete Linear Systems with Delays 571
Remark E.6. As the strictly positive vector λ we may choose the equilibrium point (E.60) since for ex=λ we have
nnAAAAAA 11))(()( 1202020 −=++−=+ −λ . (E.66)
Theorem E.19. The positive system (E.54) is asymptotically stable if and only if both matrices
nIA −1 , 20 AA + (E.67)
are Hurwitz Metzler matrices, where =
=q
k
kAA0
00 , =
=q
k
kAA0
11 , =
=q
k
kAA0
22 .
Proof is given in [55]. To check the stability of the positive system (E.54) the following procedure can
be used.
Procedure E.2
Step 1. Check if at least one diagonal entry of the matrix nnq
k
kA ×+
=ℜ∈
01 is
equal or greater than 1. If this holds then the positive system with delays (E.54) is unstable.
Step 2. Using Theorem E.4 check if the matrix n
q
k
k IA −=0
1 is Hurwitz Metzler
matrix. If not the positive system with delays (E.54) is unstable.
Step 3. Using Theorem E.4 check if the matrix ==
++q
k
kq
k
k AAA0
01
202 is
Hurwitz Metzler matrix. If yes then the positive system with delays (E.54) is asymptotically stable.
Example E.5. Consider the positive system (E.54) with the matrices
.2.001.0
15.01.0,
09.005.0
05.01.0,
01.001.0
02.001.0
,95.005.0
06.0,
3.01.0
2.04.0,
4.01.0
2.03.0
12
11
10
02
01
00
=
=
=
−
−=
=
=
AAA
AAA
(E.68)
By Theorem E.14 the system is positive since nMA ∈02 ,
2212
11
01
10
00 ,,,, ×
+ℜ∈AAAAA and 2202
01
00 115.0055.0
01.007.0 ×+ℜ∈
=+ AAA .
572 E Positive 2D Continuous-Discrete Linear Systems
Using Procedure E.2 we obtain the following
Step 1. All diagonal entries of the matrix 11
01 AAA += are less than 1.
Step 2. The matrix
−
−=−+
61.015.0
25.05.02
11
01 IAA is Hurwitz Metzler matrix
since the coefficients of the polynomial
27.011.161.015.0
25.05.0]det[ 2
211
012 ++=
+−−+
=+−− sss
sIAAsI
(E69)
are positive.
Step 3. The matrix
−
−=+++
34.017.0
37.019.012
02
10
00 AAAA is also Hurwitz
since the coefficients of the polynomial
0017.053.034.017.0
37.019.0]det[ 21
202
10
002 ++=
+−−+
=−−−− sss
sAAAAsI
(E.70)
are positive.
By Theorem E.19 the positive system (E.54) with (E.68) is asymptotically stable. The polynomial (E.64) for the positive system has the form
0017.026.019.037.027.017.125.111.1
)]1)(()()1(det[
222222
12
02
11
01
10
002
++++++++=
++−+−−−+
zszsszszzszs
zAAsAAAAzsI
(E.71)
and its coefficients are positive. Therefore, by Theorem E.18 the positive system (E.54) with (E.68) is asymptotically stable.
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Subject Index
A asymptotic stability
continuous-discrete systems Fornaisni-Marchesini 562 Roesser model 566
continuous-time systems 111, 250, 525 Descriptor 376 fractional order 354
discrete-time systems 67 descriptor 436 fractional order 520
B
boundary conditions 378, 430, 570
C
canonical form 15, 308, 332
controller form 7, 28 Frobenius form 352, 528 Jordan form 533 observer form 32
Caputo continuous-discrete systems
2D general model 418 positive 447
Fornaisni-Marchesini model 562 Roesser model 427
positive 436 continuous-time systems
descriptor 436
minimal 59 positive 280
fractional order 520 positive 545 with delays 369
fractional order descriptor 395 positive 436
positive 520, 523 cone 59, 82, 83 cyclic
matrices 148, 513 realization 548, 549
D
decomposition 399, 406 descriptor 51, 57, 376 diagonal 105, 201, 219, 414, 469, 500,
564, 572 different order 519 discrete-time systems 67
cone 82 descriptor 59
fractional order 516 positive 67
fractional order 516 with delays 123
dual system 510
E
electrical circuits 264, 266, 401 elementary operations 573 equivalent
equations 485
588 Subject Index
system 318, 512 eigenvalues 15, 42, 134, 201, 221, 225,
229, 243, 245, 375, 501, 502, 503, 508, 509, 563, 570
F
function gamma 521 Mittage-Leffler 521
fractional order 354 derivative 521 difference 511
G
general model 420
H
Hurwitz matrix 241, 563, 564, 565, 572
I
impulse Dirac 493, 522 function 396, 421 responses 115, 338, 493, 494, 497 unite 513
inverse transform 380
J
Jordan form 544 block 540
L
left elementary operations 514 half 570 prime 514, 515
lower rows 392 triangular 106, 200, 209, 213, 500
M
matrix diagonal 469 Metzler 98, 101, 103, 106, 107, 109,
110, 111, 113, 149, 196, 200, 497, 522
monomial 131, 196, 198, 218, 288, 413, 514
Schur 230 spectral radius 502, 517
monomial column 506
N
nilpotent matrix 2, 42, 298, 383, 399
O
off-diagonal 103, 106, 522 orthant 365, 497
P
positivity
continuous-discrete system 2D general model 418 Roesser model 429
continuous-time systems descriptor 276, 298 external 493 fractional with delays 364 internal 497 with delays 144, 152
discrete-time systems descriptor 275, 306 external 495 fractional 338 internal 497 with delays 123, 160, 185
R
reachability 82, 506 realization problem 1–5, 7, 9, 10, 12,
13, 15–22, 24, 25, 27–29 reduction 89
Subject Index 589
S
solution of matrix equation 484 of state equation 364, 383, 396, 399,
418, 497, 498, 512, 519, 521, 556 state equations 562
continuous-discrete systems Fornasini-Marchesini 562, 566 general model 420 Roesser model 566 with delays 505
continuous-time systems descriptor 44
with delays 364, 370 generalized 1
canonical 1 fractional orders 340, 353
descriptor 376, 403 with delays 364
Weierstrass canonical 38, 40 with delays 144, 152
discrete-time systems 67, 68, 81, 122 descriptor 258, 269
with delays 318, 366 fractional 364, 376
descriptor 436 with delays 144, 318, 160, 171
T theorem
Cayley-Hamilton 82 convolution 380, 396, 397
transfer function continuous-discrete 436
with delays 505 continuous-time systems 38, 44
descriptor 38, 43, 51, 59, 276, 298, 306, 317 fractional orders 370
with delays 364 with delays 144 discrete-time systems 67, 68, 81, 122 descriptor 295
with delays 123 fractional order 340
descriptor 376 with delays 170
transform Laplace 7, 36, 166, 354, 363, 364,
365, 396, 430, 471
U
unimodular 129, 306, 550, 551, 483 unstable 100, 121, 219, 228, 518, 565,
567, 568, 571
W
Weierstrass canonical form 38
Z
zero column 263 matrix 128 row 404
zeros 101, 114, 200, 211–213, 221–224, 227, 245, 252, 256, 263, 373