A GENERAL SOLUTION TO THE OUTPUT-ZEROING
PROBLEM FOR DISCRETE-TIME MIMO LTI SYSTEMS
Signal Processing, Systems Modelling and Control
Jerzy Tokarzewski
Military University of Technology, Kaliskiego 2, 00-908 Warsaw, Poland
Lech Sokalski and Andrzej Muszyński
Automotive Industry Institute, Jagiellońska 55, 03-301Warsaw, Poland
Keywords: Linear multivariable systems, State-space methods, Output-zeroing problem, Invariant zeros.
Abstract: The problem of zeroing the output in an arbitrar
y linear discrete-time system S(A,B,C,D) with a
nonvanishing transfer-function matrix is discussed and necessary conditions for output-zeroing inputs are
formulated. All possible real-valued inputs and real initial conditions which produce the identically zero
system response are characterized. Strictly proper and proper systems are discussed separately.
1 INTRODUCTION
The problem of zeroing the system output is strictly
related to the notion of multivariable zeros. These
zeros, however, are defined in many, not necessarily
equivalent, ways (for a survey of these definitions
see MacFarlane and Karcanias, 1976; Schrader and
Sain, 1989; see also Bourles and Fliess, 1997). The
most commonly used definition employs the Smith
canonical form of the system (Rosenbrock) matrix
and determines these zeros (which will be called in
the sequel the Smith zeros) as the roots of diagonal
(invariant) polynomials of the Smith form
(Rosenbrock, 1970, 1973). The output-zeroing
problem for continuous-time systems in relationship
with the Smith zeros was studied, under certain
simplifying assumptions concerning the systems
considered, in (MacFarlane and Karcanias, 1976),
where the notions of state-zero and input-zero
directions were introduced, and was interpreted
geometrically in (Isidori, 1995, pp. 164, 296). A
more detailed analysis indicates that for
characterizing the output-zeroing problem the notion
of Smith zeros is too narrow (Tokarzewski, 2002;
Tokarzewski and Sokalski, 2004). However,
extending in a natural way the concept of the Smith
zeros, the above difficulty can be overcomed. Such
an extension is based on the definition of invariant
zeros which employs the system matrix and zero
directions with nonzero state-zero directions (see
Tokarzewski, 2002; Tokarzewski and Sokalski,
2004). Because to each invariant zero we can assign
a real initial condition and a real-valued input which
produce the zero output, the invariant zeros can be
easily interpreted in the context of the output-
zeroing problem. Of course, since each Smith zero is
also an invariant zero, this interpretation remains
valid also for Smith zeros.
Taking into account the ab
ove concept of invariant
zeros, we can state the following question: find a
state-space characterization of the output-zeroing
problem (at least in the form of necessary
conditions) which determines in a simple manner all
the possible real-valued inputs and real initial
conditions which produce the identically zero
system response. For continuous-time systems the
question was discussed in (Tokarzewski, 2002) and
for the discrete-time case it was outlined for square
decouplable systems in (Tokarzewski, 2000).
2 PRELIMINARY RESULTS
Consider a discrete-time system S(A,B,C,D) with m
inputs and r outputs
137
Tokarzewski J., Sokalski L. and Muszy
´
nski A. (2005).
A GENERAL SOLUTION TO THE OUTPUT-ZEROING PROBLEM FOR DISCRETE-TIME MIMO LTI SYSTEMS - Signal Processing, Systems Modelling
and Control.
In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Signal Processing, Systems Modeling and
Control, pages 137-142
DOI: 10.5220/0001162101370142
Copyright
c
 SciTePress

)k()k()k(
)k()k()1k(
DuCxy
BuAxx
+=
+=+
, , (1)
,...}2,1,0{k =∈ N
where
and
are real matrices of appropriate
dimensions. By
U we denote the set of admissible
inputs which consists of all sequences
.
rmn
RRR ∈∈∈ )k(,)k(,)k( yux
D0C0BA ,,, ≠≠
m
RN →:(.)u
The point of departure for our discussion is the
following formulation of the output-zeroing problem
(Isidori, 1995): find all pairs
, consisting
of an initial state
and an admissible input
, such that the corresponding output of
(1) is identically zero for all
. Any nontrivial
pair (i.e., such that
or ) of this
kind is called an output-zeroing input. Note that in
each output-zeroing input
,
should be understood simply as an open-loop control
signal which, when applied to (1) exactly at
, yields for all .
))k(,(
o
o
ux
n
R∈
o
x
)k(
o
u )k(y
N∈k
0x ≠
o
0u ≠)k(
o
))k(,(
o
o
ux )k(
o
u
o
)0( xx = 0y =)k( N∈k
Moreover, we consider the following definition of
invariant zeros (Tokarzewski, 2000): a complex
number
is an invariant zero of (1) if and only if
(iff) there exist vectors
(state-zero
direction) and
(input-zero direction) such
that
λ
n
C∈≠
o
x0
m
C∈g
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
λ
0
0
g
x
P
o
)( , (1a)
where
denotes the system
matrix. Transmission zeros of (1) are defined as
invariant zeros of its minimal (i.e., reachable and
observable) subsystem.
⎥
⎦
⎤
⎢
⎣
⎡
−−
=
DC
BAI
P
z
)z(
The set of all invariant zeros of (1) is denoted by .
I
Z
System (1) is called degenerate iff
I
is infinite.
Otherwise, the system is said to be nondegenerate.
The set of all Smith zeros of (1) we denote by
.
Z
S
Z
Recall (Tokarzewski and Sokalski, 2004) that
; moreover, (1) is nondegenerate iff
, and (1) is degenerate iff . Recall
also that in case of nondegeneracy the Smith and
invariant zeros are exactly the same objects
(including multiplicities).
IS
ZZ ⊆
SI
ZZ =
C=
I
Z
The same symbol
is used to denote the state-
zero direction in the definition of invariant zeros and
the initial state in the definition of output-zeroing
inputs. The state-zero direction
must be a
nonzero vector (real or complex). Otherwise, the
definition of invariant zeros becomes senseless (for
any system (1) each complex number may serve as
an invariant zero). In other words, in the definition
of invariant zeros the condition
can not be
omitted.
o
x
o
x
0x ≠
o
According to the formulation of the output-zeroing
problem, the initial state must be a real vector
(but not necessarily nonzero). If the state-zero
direction
is a complex vector, then it gives two
initial states
and (and, of course, at
least one of these initial states must be a nonzero
vector).
o
x
o
x
o
Re x
o
Im x
Recall (Tokarzewski, 2000, Remark 1) that if
ϕ
λ=λ
j
e
is an invariant zero of (1), i.e., a triple
satisfies (1a), then this triple generates
two output-zeroing inputs. Namely, the pair
, where
g0x ,,
o
≠λ
))k(,(Re
o
o
ux
N∈ϕ−ϕλ= k),ksinImkcos(Re)k(
k
o
ggu
(1b)
is an output-zeroing input (and produces the solution
of the state equation of (1) of the form
)ksinImkcos(Re)k(
oo
k
ϕ−ϕλ= xxx
).
Similarly, the pair
, where
))k(,(Im
o
o
ux
N∈ϕ+ϕλ= k),kcosImksin(Re)k(
k
o
ggu
(1c)
is an output-zeroing input (and produces the solution
of the state equation of (1) of the form
)kcosImksin(Re)k(
oo
k
ϕ+ϕλ= xxx
).
We denote by
the Moore-Penrose pseudo-
inverse of matrix M (Ben-Israel and Greville, 2002).
Recall (Gantmacher, 1988) that for a given rxm real
M of rank p, a factorization
with an
rxp
and a pxm is called the skeleton
factorization of M. Then
is uniquely
determined as
, where
and .
Moreover,
+
M
21
MMM =
1
M
2
M
+
M
+++
=
12
MMM
T
1
1
1
T
11
)( MMMM
−+
=
1T
22
T
22
)(
−+
= MMMM
MMMM
+
=
and .
+++
= MMMM
Consider the equation
, , where
M is a rxm real and constant matrix of rank
and
is a fixed sequence, and suppose that
)k()k( bMz = N∈k
p
r
RN →:(.)b
ICINCO 2005 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL
138

this equation is solvable in the class of all sequences
(i.e., there is at least one solution).
Then any solution can be expressed as
, where and
is a solution of the equation
m
RN →:(.)z
)k()k()k(
ho
zzz +=
∗
)k()k(
o
bMz
+∗
=
)k(
h
z 0Mz
=
)k(
.
3 MAIN RESULTS
3.1 Proper Systems
)( 0D ≠
A general characterization of output-zeroing inputs
and the corresponding solutions is given in the
following result.
Proposition 1 Let
be an output-
zeroing input for a proper system (1) and let
denote the corresponding solution. Then
and , , has the
form
))k(,(
o
o
ux
)k(
o
x
CDDIx )(Ker
o +
−∈
r
)k(
o
u N∈k
∑
+−−
+−−=
−
=
−−++
++
1k
0
hh
1k
ok
o
)k()]()([
)()k(
l
l
l uBuCBDACD
xCBDACDu
(2)
for some sequence
∈
(.)
h
u
U satisfying
0Du
=
)k(
h
for all
, and , , has the form
N∈k )k(
o
x N∈k
∑
−+
−=
−
=
−−+
+
1k
0
h
1k
ok
o
)()(
)()k(
l
l
lBuCBDA
xCBDAx
. (3)
Moreover,
for all
CDDIx )(Ker)k(
o
+
−∈
r
N
∈
k
.
Remark 1 Proposition 1 does not tell us whether
the output-zeroing inputs exist. However, if the set
of invariant zeros is nonempty, for each such zero
there exists an output-zeroing input (see (1b) and
(1c)) which in turn may be characterized as in
Proposition 1.
Proposition 2 Let
be an output-
zeroing input for a proper system (1) and let
denote the corresponding solution. Then
))k(,(
o
o
ux
)k(
o
x
(i) if
, then
.
0DDIB =−
+
)(
m
ok
o
)()k( xCBDAx
+
−=
Moreover, the pair
, where
, is also output-
zeroing and yields
.
))k(,(
o
o
∗
ux
ok
o
)()k( xCBDACDu
++∗
−−=
ok
o
)()k( xCBDAx
+
−=
(ii) if D has full column rank, then
and
.
ok
o
)()k( xCBDACDu
++
−−=
ok
o
)()k( xCBDAx
+
−=
Remark 2 The assumption
does not imply in general that
for all
0DDIB =−
+
)(
m
)k()k(
oo
uu =
∗
N
∈
k
. It implies, however, that and
applied at the initial state
affect the state
equation of (1) in the same way. This follows from
the relation
.
)k(
o
u )k(
o
∗
u
o
x
0BuBu =−
∗
)k()k(
oo
When D has full row rank, the necessary condition
given by Proposition 1 becomes also sufficient.
Proposition 3 In (1) let D have full row rank.
Then
is an output-zeroing input iff
has the form (2), where and
is an element of U satisfying
for all
))k(,(
o
o
ux
)k(
o
u
n
R∈
o
x (.)
h
u
0Du =)k(
h
N
∈
k
. Moreover, the solution corresponding to
has the form (3).
))k(,(
o
o
ux
A more detailed characterization of the output-
zeroing problem than that obtained in Proposition 2
(ii) is given by the following result.
Proposition 4 In (1) let D have full column rank.
Then
is an output-zeroing iff
))k(,(
o
o
ux
})(){(Ker:S
1
0
o l
n
l
r
cl
CBDACDDIx
D
++
−
=
−−=∈
∩
(4
and
)
xCBDACD
++
−−=
. (5)
Moreover, the corresponding so
+
−=
(6)
ok
o
)()k(u
lution equals
ok
o
)()k( xCBDAx
and is contained in the subspace
(4), i.e.,
cl
for all
cl
D
S
D
x S)k(
o
∈ N
∈
k
.
Remark 3 Any proper system (1) can be
transformed, by introducing an appropriate pre-
compensator, into a proper system in which the first
A GENERAL SOLUTION TO THE OUTPUT-ZEROING PROBLEM FOR DISCRETE-TIME MIMO LTI SYSTEMS -
Signal Processing, Systems Modelling and Control
139

nonzero Markov parameter has full column rank. In
fact, suppose that
mp <
=
Drank
. Let
21
DDD
=
be a skeleton factori ducin
compensator
T
2
D
to (1), we get a sytem with the
first nonzero arkov parameter
T
DD
of full
column rank. Finally, by introdu into a
reachable system (1) the precompensator
T
2
D
,
reachability may be lost.
zation of D. Intro g the pre-
M
2
cing
or a proper system (1) we denote by F
),,,( DCBA
∗
=
r,
is the
rzewski, 2002, p. 195))
emma 1 (Tokarzewski, 2002, p.188) Consider a
, (7)
e., the system has no invariant zeros iff its maximal
roposition 5 In a proper system (1) let
inpu
and
Moreover, the corresponding to su
emark 4 It is important to note that Lemma 1
the maximal output-nulling
variant subspace (Aling and
Schumache 1984). Recall also that if
))k(,(
o
o
ux
is an output-zeroing input for (1) and
x
corresponding solution,
)()k(
∗
∈=
for
all
N∈k
(see e.g. (Toka .
controlled in
)k(
o
,,,
o
DCBAx
L
proper system (1). Then
}{),,,( 0DCBA =⇔∅=
∗
=
I
Z
i.
output-nulling controlled invariant subspace is
trivial.
P
∅=
I
Z
.
Then a pair
))k((
o
o
ux
is an output-zeroing t
for (1) iff
DBu KerKer)k(
for
all
N∈k
. ch a
pair ion of the state equation in (1) equals
0x =)k(
o
for all
N∈k
.
,
0x =
o
o
∩∈
solut
R
and Proposition 5 are not valid if we replace the
assumption
∅=
I
Z
with
∅=
S
Z
(see Example 2).
3.2 Strictly Proper Systems
)( 0D
=
If , then the first nonzero Markov parameter
(1) is
0D =
of denoted by
BCA
ν
, where 10
−
≤
ν
≤ n
(i.e.,
CACB ==
−ν
...
and
Defin rzewski a i,
2004)
=K :
0B =
1
e the matrix (comp. Toka nd Sokalsk
CABCABI )(
. (8)
As a necessary condition f
following
-
r ictly proper
0BCA ≠
ν
).
ν+ν
ν
−
or a pair
))k(,(
o
o
ux
to
be an output-zeroing input we have the .
roposition 6 Let
))k(,(
o
ux
be an output
P
o
zeroing input fo a str system (1) and let
)k(
o
x
denote the corresponding solution. Then
∩
ν
0
S
l
x
+
∑
−
+−=
−
=
−−
ν
+ν+ν
ν
+ν+ν
l
l
l
, (9)
for some
=
ν
= Ker:
l
CA
and
)k(
o
u
has the form
∈
o
)k(
)]()([)(
)()()k(
h
1k
0
h
1k1
ok1
o
u
BuAKCABCA
xAKCABCAu
∈
(.)
h
u
U satisfying for
0BuCA =
ν
)k(
h
all
N
∈
k
, and
)k(
o
has the fo
1k
−
x
rm
0
ok
o
l
l
l
BuAKxAK
−−
=
νν
∑
+=
. (10)
Moreover,
is entirely contained
, i.e.,
)()()()k(
h
1k
x
)k(
o
x
in the
subspace
ν
S
ν
∈
S)k(
o
x
for all
N∈k
.
Remark 5 Note er the a ti that und ssump ons of
Proposition 6 the input (9) (at any
∈
(.)
h
u
U
satisfying
0BuCA =
ν
)k(
h
for all
N∈k
d
to the syste y initial s
n
R∈)0(
yields the solution of the state equation of
)k())0(()k(
o
ok
xxxAx +−=
, where
)k(
o
x
is as
em outp equals
))0(()k(
ok
xxCAy −=
. In particular, if A is stable,
0y →)k(
as
∞→k
.
Rema osition ot te h
) applie
m at an arbitrar tate
x
the form
rk 6 Prop 6 does n ll us w ether
Let
be an output-
r ictly proper
(i) i
in (10), and the syst ut
then
)k()k(
o
xx →
and
the output-zeroing inputs exist. However, if the set
of invariant zeros is nonempty, for each such zero
there exists an output-zeroing input (see (1b) and
(1c)) which in turn may be characterized as in
Proposition 6.
roposition 7 P
))k(,(
o
o
ux
zeroing input fo a str system (1) and let
)k(
o
x
denote the corresponding solution. Then
f
0BK
=
ν
, then
ok
o
)()k( xAKx
ν
=
.
ICINCO 2005 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL
140

Moreover, at
the pair , where
, is also
output-zeroing and yields the solution
.
0BK =
ν
))k(,(
o
o ∗
ux
ok1
o
)()(:)k( xAKCABCAu
ν
+ν+ν∗
−=
ok
o
)()k( xAKx
ν
=
(ii) if
has full column rank, then
and
.
BCA
ν
ok1
o
)()()k( xAKCABCAu
ν
+ν+ν
−=
ok
o
)()k( xAKx
ν
=
Remark 7 The assumption
does not
imply in general the equality
for all
, although it implies . The
reason behind this becomes clear if we consider the
relations
0BK =
ν
)k()k(
oo
uu =
∗
N∈k )k()k(
o
xx ≡
=−−=−
ν
∗
)k()k()()k()k(
oooo
uBxAIKuBuB
)k()1k()k(
ooo
uBKxAxK
νν
−
=+−
.
Thus, at
, although in general
, both these inputs applied at the
initial state
affect the state equation of (1)
0BK =
ν
)k()k(
oo
uu ≠
∗
o
)0( xx =
in exactly the same way.
Proposition 8 In a strictly proper system (1) let
have full row rank. Then is an
output-zeroing input iff
and
is as in (9) with U satisfying
for all . Moreover, the
corresponding solution
has the form (10) and
is entirely contained in the subspace
, i.e.,
for all .
BCA
ν
))k(,(
o
o
ux
∩
ν
=
ν
=∈
0
o
KerS
l
l
CAx
)k(
o
u ∈(.)
h
u
BCAu
ν
∈ Ker)k(
h
N∈k
)k(
o
x
ν
S
ν
∈S)k(
o
x N∈k
Proposition 9 In a strictly proper system (1) let
have full column rank. Then a pair
is an output-zeroing input iff
BCA
ν
))k(,(
o
o
ux
l
n
l
cl
)(Ker:S
1
0
o
AKCx
ν
−
=
ν
=∈
∩
(11)
and
ok1
o
)()()k( xAKCABCAu
ν
+ν+ν
−=
, (12)
N∈k
Moreover, the solution of the state equation
corresponding to
has the form
))k(,(
o
o
ux
ok
o
)()k( xAKx
ν
=
,
N
∈
k
, (13)
and is entirely contained in the subspace
, i.e.,
for all
cl
ν
S
cl
ν
∈S)k(
o
x
N
∈
k
.
Remark 8 Any strictly proper system (1) with
nonvanishing transfer-function matrix can be
transformed, by introducing an appropriate pre-
compensator, into a strictly proper system in which
the first nonzero Markov parameter has full column
rank. In fact, suppose that
. Let
be a skeleton factorization.
Introducing to (1) the precompensator
, we get a
system with the first nonzero Markov parameter
of full column rank. By introducing to a
reachable system (1) the precompensator
,
reachability may be lost.
mp <=
ν
BCArank
21
HHBCA =
ν
T
2
H
T
2
BHCA
ν
T
2
H
For a strictly proper system (1) we denote by
the maximal output-nulling controlled
invariant subspace (Basile and Marro, 1992;
Wonham, 1979; Sontag, 1990). Recall also that if
is an output-zeroing input for (1) and
is the corresponding solution, then
for all
),,( CBA
∗
=
))k(,(
o
o
ux
)k(
o
x
),,()k(
o
CBAx
∗
∈= N∈k
Lemma 2 (Tokarzewski, 2002, p.168) Consider a
strictly proper system (1). Then
}{),,( 0CBA =⇔∅=
∗
=
I
Z
, (14)
i.e., the system has no invariant zeros iff its maximal
output-nulling controlled invariant subspace is
trivial.
Proposition 10 In a strictly proper system (1) let
. Then a pair is an output-
zeroing input for (1) iff
and
for all
∅=
I
Z ))k(,(
o
o
ux
0x =
o
Bu Ker)k(
o
∈
N
∈
k
. Moreover, the corresponding to such
a pair solution of the state equation in (1) equals
0x
=
)k(
o
for all
N
∈
k
.
Remark 9 It is important to note that Lemma 2
and Proposition 10 are not valid if we replace the
assumption
with (see Example 1).
∅=
I
Z ∅=
S
Z
A GENERAL SOLUTION TO THE OUTPUT-ZEROING PROBLEM FOR DISCRETE-TIME MIMO LTI SYSTEMS -
Signal Processing, Systems Modelling and Control
141

4 EXAMPLES
Example 1 In (1) let
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
−−
=
010
001
,
001
010
101
,
100
03/20
103/1
C
BA
.
The system has no Smith zeros; on the other hand
(see Tokarzewski and Sokalski, 2004, Proposition
9), it is degenerate (i.e.,
). Since has full
row rank, all output-zeroing inputs are as in
Proposition 8. Note that
, i.e.,
the maximal output-nulling controlled invariant
subspace is nontrivial (comp. Remark 9).
C=
I
Z CB
CCBA Ker),,( =
∗
=
Example 2 In (1) let
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
−−
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−−−
=
000
101
,
110
102
,
100
010
001
,
010
000
12/12
DC
BA
.
The system has no Smith zeros; on the other hand
(see Tokarzewski, 2002, p.188), it is degenerate and
is non-
trivial (comp. Remark 4).
}0xx:{),,,(
32
3
=+∈=
∗
RxDCBA=
5 CONCLUDING REMARKS
In this paper we presented necessary conditions for
output-zeroing inputs and the corresponding
solutions (Propositions 1 and 6) for a general class
of linear discrete-time systems described by the
state-space model (1). It is shown that if the first
nonzero Markov parameter has full row rank, the
necessary conditions become also sufficient
(Propositions 3 and 8). Necessary and sufficient
conditions for output-zeroing inputs for systems
with the first nonzero Markov parameter of full
column rank are given in Propositions 4 and 9.
Finally, necessary and sufficient conditions for
output-zeroing inputs under the assumption that the
set of invariant zeros is empty are presented in
Propositions 5 and 10.
A more detailed characterization of the output-
zeroing problem can be obtained by using singular
value decomposition of the first nonzero Markov
parameter.
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