IEEE Circuits and Systems Magazine - Q3 2022 - 19

■ SISO system:
()
Dt tt tt
() () ()tt tt t
a
xA xB v
NN NN
=+
## #
vector
=+
##
vector
vector
()
yC xD v
NN
■ MIMO system with nv
matrix
Dt tt tt
()
a
xA xB v
NN N
()
=+
## ##
1 vector
tt tt t
: : : :
matrix 11nv# vector
yC xD v
nN N
=+
## #
() ()
y
vector nn
yv matrix
The above equations represent the standard form of
expressing a linear system by means of state variables,
which are the elements of the state ().tx A system is time
invariant if matrices A, B, C, and D are constant. This
is the situation assumed in the following. Therefore, a
given LTIS has a state-space representation in the form
Dt ttxAxBv=+a
() () ()
yCxDv=+ () () ().tt t
s ,ka
Define the diagonal matrix diag sa
=12 f ,, , .
kN
diag ss ssXAXBV=+a
YCXDV=+
^h
From (94),
XA BV1
() () ()ss s=-a
-
diag^h
and replacing this in (95) yields
()
V
Y
aak
=
()
s
s
diag sa
k ,, ,n12 f=
(96)
The LT of (92)-(93) is
() ()
()
ss s
(92)
(93)
^h with elements
()
() ().
(94)
(95)
() ()
()
()
inputs and ny
Nn matrix n
() ()
: : ::
11 scalar scalar
vv vector
D D D D
() ()
outputs:
1
(90)
(91)
Therefore, there exists a state transition operator,
U (, ),t0
verifying
x tt00x$
()=U(, )( ),
that is expressed by the inverse LT
(,).Lts s
U =-a
0diagdiag
-1 " 6 ^^ -hh@A -
a 1
1
$
,
(101)
and assume that the input is null for t 02 Using the re.
matrix
11vector scalar
D D D D
() ()
()
(88)
(89)
sults introduced in (49), we can write
()
diag ss ss0Xx AX().
a -=a^^
hh
(100)
diag
1
()
(102)
There is a closed form for such inverse, but we do not
present it, since it is a bit involved [40], [42] and it is not
necessary in the follow-up. In the commensurate case,
we can write
U =-aa-
(,
) Lts s -1 I,
$
and
U (, ) Lts0 --a-1
=
1 Ann
'/
3
3
U (, ) / Ak
k=0
0 nh =
h
k ()a ka +1 n
n!
1
(104)
that assume different forms according to the used LT. In
the NLT case, we obtain from (30) the result
f nh
(),
while in the bilinear case, we get
(, )( ).
U 0 nh = / ` j
3
Ak
k=0
=- +CA BD1
6 ^h @h
2
,
(97)
which
is the TF matrix. If the derivative orders are equal,
,
rable state-space representation in which
diag ss ,I=a
^h
where I is the identity matrix.
Remark IV.1. To simplify, we use the generic designation
LT, without making a clear distinction between the
different transforms, namely in the bilinear case, where
()/(
s zz )1111
=- +-- is implicit.
C. State Transition Operator
Let us come back to the dynamic equation
Dt ttxAxBu=+a
THIRD QUARTER 2022
() () ()
(99)
a
01 ,1 #a we obtain the commensu(98)
h
3
U
0 t = / Ak
k=0
ka+1
} f nh
--1
n
ka
(105)
1 " 6 IA@ 1
(103)
(106)
If we use a causal CT system, the inverse LT is given by
(, )
t
ka
C()
ka +1
f t
(),
.
(107)
that is the multidimensional MLF [43]. This is the result
we obtain from (105) or (106) when
with
k aa++
= h
aa 1
n
tn ,h= we can write
!
()
k +1 n
=
and
limh
h " 0
k ()a ka +1 n
n!
t
=
C ka +1
volved, but yields an identical result.
With (, ),t0U
operator, (, )txU
()
.
The case of expression (106) is somehow more inwe
can define the general state transition
relating the states at two instants x and :t
IEEE CIRCUITS AND SYSTEMS MAGAZINE
19
ka
a
a
CC
C
n +1
k knC()
C()
(/ )
() (/
kth
kthh
++
++
ka
11
1
) ()C ka +1 8 B
.
h
ka
h
t
ka
h 0 " For the first,
IEEE Circuits and Systems Magazine - Q3 2022

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