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BlockForm Control of Linear Timelnvariant
DiscreteTime Systems Defined over Commutative Rings
E. \M. Kamen
School of Elec'tri,cal and Computer Engi,neering
Georgia lnstitute of Technology
Atlanta, Georgia 303320250
Submitted by A. C. Antoulas
ABSTRACT
A new approach based on a blockinput, blockoutput state model is developed for
the study of li"ear timeinvariant discretetime systems whose coefficients belong_ to 1
commutative ting with 1. It is well larown that such systems arise in the study of
various classes of complex linear systems including systems depending on parameters
and multidimensional systems. By timecompressing the blockinput state representation, new results are obtained on the construction of a memoryless blockform state
feedback control law that ields a type of assignability and/or deadbeat control. These
results are then duali zed to ield results on a new type of state observer based on a
block of output measurements. The observer and state feedback controller are then
combined to yield new results on inputoutput regulation and setpoint trach"g Pt
systems defined over an arbitrary commutative ring. In the last part of the paper, the
blockinput form is utilized to study the stabilization of systems deftned over a normed
algebra.
1.
INTRODUCTION
?
i
t
:
i
I
g
In addition to the welldeveloped the ory of linear timeinvariant systems
based on linear equations with coefficients in the field R of real numbers,
there is a large b"dy of work on linear timeinvariant systems whose coefficients belong to a commutative .itrg R [see Sontag (1976, 1985), Brewer et al.
*A
preliminary version of this paper was presented at the IMA Workshop on Linear Ngebra
for Control Theory, Minneapolis, june 1992.
LII{EAR ALGEBRA AI,{D ITS APPLICATIOI:{S 205_206:805_829
@ Elsevier Science Inc., 1994
655 Avenue of the Americas, New York, NY 10010
(1994)
oo2437e5
/e4/
805
$7.00
806
E. W. KAMEN
(1986), and Kamen (fggt)]. In the discretetime case, an minput,
poutput
ndimensional linear timeinvariant system defined over a commutative *"g
R with multiplicative identity I is given by the state model
*(k + t) : Ax(k) * Bu(k),
vG)
where
ft is the discretetime
:
(r.1)
C*(k),
index, A,
(
B,C
are
1.2)
nXn, nXm, pXn
matrices over R, and the input z(ft), output y(k), andstate r(ft) are column
vectors over R.
of systems of the form (1.1), (I.2) that arise in applications
.Examples
integer systems with R : Z : the ring of integers, po*rotrtz"d
lncluje
famllles of systems with R a ring of realvalueJ functions deffned on some
parameter set such as a subset of Ndimensional Euclidean space RN, and
maltidi.rnensional systems with R a convolution ring of realvaiued functions
deftned on the sfold cartesian product z x z x ... x Z for some integer s
[see Sontag (1976, 1985) and Kamen (fgsf) for details].
one of the topics t}at has been extensively studied in the literature on
systems over rings is state feedback control using a mem.oryless (or static)
controller given by
"(k): Fx(k)
or a dynamic controller of dimension r gpven by
"(k): Fx(ft)
(
+ Gu(k),
1.3)
(L.4)
where D(k) is the rdimensional state vector of the feedback controller given
by
"(k + l)
It
(t.3)(1.5), F, G, H, J
ring R.
are
:
Ho(k) + I*(k)
m x n, m x r,
(
1.5)
r x r, r x n mat*;1ces over the
Part of the focus of the past work has been on the problem
of
assi,gnabl,t
itA by state feedback; in other words, the control objective is to "assi[n" (in
some sense)
lh", dyt?*ics of the closedloop system by using stat" f"edback
of the form (t.g) or (1.4), (1.5). In particular, in the case of the memoryless
state feedback control (t.g), we can ask whether or not we have cofficient
assi,gnabtlltythat is, glven any elements as, a1,..., an_L belongi"gi" the
tittg R, whether there is a feedback gain matrix F over R iuch that with the
LIhJEAR TIMEINVARIANT DISCRETE.TIME SYSTEMS
807
control (1.3), the resulting closedloop characteristic polynomial det(zl*  A
+ BF) is equal to zn * anLznr + "' *arz * ao. Here ln is the n X n
identity matrix.
For systems over a field, it is well known that coefficient assignability is
equivalent to reachability of the gtven system, which in turn is equivalent to
tB]. Howright invertibility of the reachability rnatrix f : lB AB "' Anever, for systems over a commutative ritrg R, in general reachability is not
sufftcient for coefftcient assignability by memoryless state feedback, where in
tB] has a right
the ring case, reachability means that f  ln AB "' Aqinverse over R for some integ er q
In the case of dynamic state feedback given by (L.4), (1.5), coefficient
assignability is deftned as follows. First, combining (1.1), (t.Z) with (L.4),
(1.5), we have the following (n + r)dimensional state equation for the
closedloop system:
l;ll I ill
:LleBF
T
'rl [;[]l
The closedloop characteristic polynomial is
A
_I
+BF BGl
zl, Hl'
(
1.6)
Then the given system (1.1), (1.2) is dynami,cally coefficient assignable if
there exists a positi'u" integer r such that the.coeffrcients of the (n + r)degree polynomial defined by (1.6) can be assigned arbitrarily by choosing F,
G,H, I,
In the work of Emre and Khargonekar (fgSZ), it was shown that arry
reachable system (1.1), (f.2) over a commutative ring R is dynamically
coefftcient assignable. However, in the general case when there are no
restrictions on the ring R, existing techniques for constructing a dynamic
state feedback controller yield controllers whose dimension r is in general on
the order of n2, which can be rather excessive even for systems whose
dimension n is not large. For certain special classes of rings, there exist
assignabilitytype results using memoryless or loworder dynamic state feedback controlleis [e.g., see Sontag (1985), Brewer (f992), and Brewer et al.
(lees)1.
In this paper a new approach based on a blockinput, blockouput state
model is developed for the study of linear timeinvariant discretetime
systems deffned over an arbitrary commutative ring R with 1. For linear
dmeinvariant and periodically timevarying discretetime systems deffned
808
E. W. KAMEN
over the reals R, results based on blockinput, blockoutput models (also
called lifted representati.ons) have been derived by a numbei of authors [e.g.,
see Khargonekar et al. (1985), powell et al. (1g87), Buescher (fggg),
Haigawara and Araki (I98S), Francis and Georgiou (19gg), Albertos (f990),
Albertos and Ortega (1989), and Lozano (l9S9)1. In the work of Kamen
(1992a, 1992b), a blockform approach is developed for the class of linear
discretetime systems with general timevarying coefftcients, and in the work
of Gnzzle and Kokotovic (rgss) and Grizzle and Moraal (rgg0), a blockform
approach is developed for a large class of nonlinear timeinvariant discretetime systems.
For linear timeinvariant discretetime systems over a commutative ring
(which includes systems over the reals R), we begin in Section 2 with th;
generation of the blockinput form of the state model. By timecompressing
this representation, new results are obtained on the construction of a memoryless blockform state feedback control law that yields a t}pe of assignability,
or that results in deadbeat control. These constructioni are duilized in
section 3 to obtain results on the assignability of the error dynamics of a new
type of state observer based on a block of output measurements. The
observer and state feedback controller are then combined in section s and 4
to yield new results on inputoutput regulation and setpoint trachng. In
section 5 of the paper, the focus is on the stabilization oT systems deffned
over normed algebra. It is shoum that the construction of stabilizing
1_
controllers can be approached in terms of the solution to a generalized lineai
quadratic formulation based on the block form.
?.
THE BLOCKINPUT FORM AND STATE FEEDBACK CONTROL
Again consider the discretetime system (1.1), (1.2) deftned over a commutative
i"g { with 1. To simplify the notation, in the following development we shall denote the system by the triple (A, B, C).
  Given a positiveinteger q ) 1, let I] denote the n X nq qstep reachabtltty matrtx deffned by
lq: ln
AB
AqrB].
in the Introduction, the system ( A, B, C) is reachable (oming
if for some integer q l, q has a right inverse over R; that is,
there exists an nry X n matrix (L)" over R such that q(q)"
 In. It follows
As noted
reachable)
from the Cuyl"yHamilton theorem for matrices over a commutative ring that
809
LINEAR TIMEINVARIANT DISCRETE.TIME SYSTEMS
the system ( A, B, C) is reachable if and only if
[see bt"*"r et al. (1986)].
Given a positive integer q
I;
has a right inverse over R
times, we have that
(/c +
q) :
AQ
*(k) + TrUn( k),
(2.1)
where Uq(k) is the mqelement blockinput vector defined by
"(k+q 1)
"(k+q2)
uq(k) :
a
u(k +
1)
"(k)
Equation (2.f) is the bLockinput form of the state equation for_the sl,stem
(i,n,C) deffned by (1.1), (t.z). t'lote that since rJn(k + 1) depends on
4(k) when q > l, we cannot view (2.1) as the state equation of a mginput
rpt"*. However, by time scaling w€ can generate a mqinput representation
as follows. The time scaling is Jccomplished by replacing k with kg in the
time signals 4(k) and r(k). If q > l, then Q(kq) and r(kg) are time
compressiont .if 4tt) and r(k), respectively. It should be noted that for
systJms over the t"'"lr (i."., R : R), tfre blockinput state equation (Z.t) and
tle idea of time scaling (2.1) can be found in the work of Powell et al. (1987),
Buescher (1988), and Albertos IfSSO).
Now replacin1 k by kq in (2.1), we obtain the state equation
*(kq + q)

Aq
*(kq) + TnUr(kq)
.
(2.2)
When g : L, (2.2) reduces to (2.1), but for q > l, (2.2) is a timecompressed
state equation of the system (A n,C). It particular, one time step in (2.2)
correspbnds to g time steps in the original state equation (f.f). Equation
(2.2) i; the qscaLed blackinput fonn of the state equation for the system
(A, B,c).
Since 4(/cq) is independent of 4((k + 1)q), it is clear that (2.2) can be
of a mqinput system over the ring R. As will be
viewed as tie siate
"q,t"tion
seen below, due to the increased number of inputs, state feedback control is
easier to carry out in terms of the state equation (Z.Z) than
(1.r).
it is in terms of
810
E. W. KAMEN
With respect to the timecompressed representation (2.2), we can consider the memoryless blockform state feedback control law given by
Un(kq)
:  Fq *(kq)
,
k:
0, I
,2,.,.,
(2.3)
where Fq rs a mq X n matrix over the ring R. Inserting the control (2.3) into
(2.2), we have that the resulting closedloop state equation on the kq time
scale is given by
x(kq +
It
q):
(en
 q F)*(kq), k:0,
1,2,...
(2.4)
should be noted that although the blockform control law (2.3)
is
memoryless (on the kq time scale), the implementation of the control (2.3)
does require that the entries of Un(kq) be stored until they are needed. Thus,
the implementation of the control (Z.g) does require memoy, but the form of
the controller dynamics is of a very special type in comparison with the
general form of the dynamic state feedback controller defined by (L.4), (1.5).
The deftnition of the block control (2.3) leads to the following fundamental concept.
I)nplNrrroN 2.L, The system (A,
assignable for some integer g
a nry X n matrix Fq over R such that
B,C) defined by (1.1), (1.2) i,
Aq lrFn :Q.
Suppose that the system ( A, B,
n X n matrix
?
C)
over R, there is an
with the state feedback control Un(kq)
system (2.4) is giver by
*(kq + q)
:
(2.5)
is gstep assignable, so
Fo
that given any
over R such that (2.5) holds. Then
:  Fq x(kq), the resulting
Q"( kq),
qstep
k

0,L,2,..,
closedloop
(2.6)
From (2.6), we see that the closedloop system's state dlmamics on the kq
if the system is gstep assignable.
We have the following necessary and sufficient condition for gstep
tim,e scale are completely assignable
assignability.
Tsnonnu 2.L. The system ( A, B, C) is qstep assignable for soffta inte71 if and only lf the system is reachable. In adrli,tion, if (A, B,C) is
ger q
LINEAR TIME.INVARIANT DISCRETETIME SYSTBMS
reachable, so that ln has a ri,ght i,noerse (I;)" oDer R, then
nstep assignable, and fo, anA n X n, m"atri,x A ooer R, Aq
satisfiedu:ith q

811
(A, B,C)
lnE, : a
OS
0s
n and
(2.7)
Proof. Suppose that (A, B,C) is reachable, so that I) has a right
inverse (I))" over R. Given Q over R, it is clear that (Z.l) is a solution to
(2.5) with q : n. Conversely, suppose that (A, B, C) is gstep assignable for
some g > 1. Then setting Q : An  1, there exists a matrix Q over R such
that Aq  lnFn : Aq  I. Thus, lrFo : I, which shows that In has a right
I
inverse over R,'and therefore the syitem is reachable.
From Theorem 2.1 we see that reachability of the system (A, g,C) is
equivalent to complete assignability on the kn time scale of the closedloop
system's state dynamics. Of course, this construction does not in general allow
for the speciftcation of arbitrary closedloop dlmamics on the original time
scale.
There is one very important case where assignability on the kg time scale
does result in a correspon&ng result on the /c time scale, and that is in
dpadbeat control. To be precise, suppose that there is an { over R such that
As lnFo:0. Then from (2.6), x(kq + q):0 for /c : 0,I,2,... and for
any inidal'state r(0), so we have deadbeat control on the kq ttme scale. But
by (2.3), U,(kq): 0 for k : L,2, ..., since x(kq\ : 0 for /c : L,2,..., and
by deftnition of U"(k), it must be true that u(k):0 for k: q,q + L,... .
Then since r(k +'f) : Ax(k) + nu(k), the state r(ft) must also be zero for
all k > g. Therefore, we also have deadbeat control on the /c time scale.
Combining the above observations with Theorem 2,1, we have the following result.
Tsnonnu 2.2. Su.ppose that the systenx ( A, B, C) is reachable so that l,
has a right irwerse (1,)' ooer R. Then with q : n and F, : (1,)'A", the
control (2.S) tu a deadbeat control on the k ttmp scab; that is, the state x(k)
of the rextlti,ng cLosedloop system i,s zero for all k > n, starting from any
ini,tial state x(0) u:ith entries in R.
By Theorem 2.2, reachability is a sufftcient condition for the existence
of
a deadbeat control on the k time scale. It is worth stressing that this result is
valid for systems over an arbitrary commutative ring R with 1.
We conclude this section with the following necessary and sufficient
condition for existence of a deadbeat control.
8L2
E. w. KAMEN
THnonnvr 2.3. There exist an i,nteger q
of the fo* (2.3) such that x(k)  0 for all k 2 q, starting from anA initial
state r(0), if and only if there exists a mq x n matrix O, oDer R such that
Aq
: ln0*
in u:hich case the control (2.3) u:ith Fn _
(2.8)
@q is
a deadbeat control on the k
tim"e scale.
Proof . If (2.8) is satisfied, it follows from the arguments glven above that
the control (2.3) with Fq _ Qq is a deadbeat control; i.e., x(k) : 0 for all
k 2 q. Conversely, suppose that there is a control of the form (Z.g) such that
x(k) : 0 for all k 2 q and for any r(0) deftned over R. Then from (2.4),
('+q lnFn)r(O)  0 for all nvectors r(0) over R. It follows that Aq
TnFr 0, and thus (2.8) is satisfted.
It should be noted that if A has an inverse
I
over R, the condition (2.8) is
equivalent to requiring that Tq has a right inverse over R, which implies that
the system (A, B,C) is reachable, So in this case, the existence of a deadbeat
control of the form (z.g) is equivalent to reachability.
3.
STATE OBSERVERS AND INPUTOUTPUT REGULATORS
In the first part of this section, w€ first consider the construction of
a
blockform state observer by "dualizittg" the results grven in the previous
section. Then the state observer is combined with a state feedback controller
to generate a type of inputoutput regulator.
Again consider the system ( A, B, C) over the commutative titrg R
deftned_by (1.1), (1.2). Given an integer q
qstep obseraablllty matrix defined by
oq
l'J,
rh"system(A,B,C)issaidtobestronglaobseraableifforsomeg>
has a left inverse over R, which is the case if and only if C), has a left inverse
over R.
LINEAR TIME.INVARIANT DISCRETETIME
SYSTEMS
8T3
With (A',C', nr) defined to be the dncal of the system (A, B,C), where
superscript T denotes the transpose operation, it is easy to see (and well
known) that the system ( A, B, C) is strongly obseruable if and only if the dual
system ( A' , C' , Br ) is reachable.
Now to be able to deftne a blockform obsenrer for the system ( A, B, C),
we first need to generate a block version of the output equation (1.2). First,
let Eq denote the pq X mq matrix deftned by
[o o ...
lo o ...
O
FIO
"ql...l
o
o
...
CB
o I
cB
CAB
I
I
L; i, ... cer'n cAq'u)
Then iterating (1.1), (t.Z)
the output equation:
q I
times, we have the following block form of
Yq(k) Qr*(/.) *
where
C)q
Enun(k),
(
3.1)
is the obsenzability matrix and Yq(k) is the pqelement blockout
put vector defined by
y
Yq(k) :
(k)
a$+1)
a$+q2)
a$+q 1)
Along with the blockoutput equation (3.1), we have the blockinput state
equation (2.1), which is repeated below for convenience:
*(k + q) : AQ*(k) + lrun(k).
(3.2)
Recall that lq is the gstep reachability matrix.
Based on the block form (3.2), (3.1) of the system
the state observer by
(A, B, C), we define
ft.(k +
q)
:
Ae
i(k) *
Lr[rn( k )
Qni(/r)  EqUr(/.)] + lrUr(k),
k
8t4
E. W. KAMEN
In (3.3), ft(k) is the estimate of the state x(k) at time k, and L o is the
n x pq observer gain matrix deftned over the rittg R. Note that (g.S) differs
from the standard form of a state obseruer in that the next estimate i(k + q)
is computed q steps ahead of the previous estimate, and the block output
vector Yq(k) is used to compute the next estimate [rather than the output
value A(k)]. Also note that in order to start the recursion deftned by (3.3), it
is necessaryto specifyinitialestimates ft,(k)for k0, 1,...,q 1.
Defining the estimnti,on error fr(k) : x(k)  ft(k) and using (3.1)(3.3),
we have that the obsenrer error dlmamics are given by
fr(k + q)
: le, 
Lqonl
*(k),
k
(3.4)
We shall say that the error dynamics giver by (g.a) are qstep assignable if for
any nXn matrixW over R, tlereis a nXpq matrix Lq over R suchthat
Aq LnQr:W.
If (3.5) holds,
(3.5)
then the error equation (3.4) becomes
fr(k + q)  wft(ft)
,
(3.6)
k
and thus, the error dlmamics are completely assignable over a qstep interval.
It is easy to see that gstep assignability of the observer error dynamics
(3.4) is dual to gstep assignability of the closedloop system dynamics (2.4) as
deftned in the previous section. More precisely, the assignability of the
observer error dlmamics is equivalent to the assignability of the closedJoop
dynamics for the dual system
following dual of Theorem 2.1.
(A',Cr,Br). In
particular, we have the
Tneonsu 3.1. The obsenser eror d4namics (3.4) are qstep assignable
for some integer q >I ff and only if the system (A,B,C) is strongly
obsercable. ln add.iti,on, if (A, B,C) i.s strongJy obsentable so that A, has a
left i,noerse (dt)t ooe, R, then the error dynimlcs are nstq assignabie, and
for any n )1 n matrixW ooer R, Aq  LrOn  W is satisfied ui,th q : n and
L,:lA" w](o,)'.
(e,
From Theorem 3.1, we see that strong observability of the
system
S, C) is equivalent to complete assignability over a nstep interval of the
observer error dlmamics. It is interesting to note that this assignability is on
LINEAR TIMEINVARIANT DISCRETE.TIME SYSTEMS
815
scale, whereas in the state feedback control problem the
assignability is on the kn time scale.
We also have the dual of Theorem 2.2.
the k time
THBonEM 3.2 .
so that On has
gain Ln
Suppose that the system ( A, B , C) ,s strongly obsertsable
iruserse (0,)' oDer R.Then usith q  n andthe obserrer
"tnft the
A"(Qn)',
x(k) : i,(k) fo, all
f(o),f(1),...,i(n 1).
obsertser (3.3)
is a
deadbeat obsertser;
that
is,
k
By Theorem 3.2, we see that strong obseruability implies that the system's
true state x(k) can be determined exactly for k 2 n starting from any initial
l). A necessary and sufftcient condition for
estimates f(0), f(1),...,i(n
the existence of a deadbeat observer of the form (3.3) is grven in
the
following result which is the dual of Theorem 2.3.
THnonnvr 3.3 . There exist an integer q
 f ( k) for all k 2 q starting from anA initial estimates
f(0), i(1), . . ., ft(q L) if and only if there exists a n x pq matrix Pn ouer R
such that
(3.3) such that x(k)
Aq
:
PnOq,
in u>hich case, the obseruer (3.3) u>i,th Lr: Pn has the deadbeat property.
In the remainder of this section it is shown that the state feedback control
law (2.3) studied in the previous section can be combined with the " qscaled
version" of the observer (3.3) to yield a type of inputoutput regulator. First,
replacing k by kq in (3.3), *" have that on the kg time scale the obsenrer is
given by
ft(kq + q)
:
Ae
+
ft(kq) + LrlYr(kq)
Lur(kq),
o
ni(kq)
k
ErUr(t q)l
(3.7)
Then in the control law (2.3), we replace x(kq) with t(kq), where i(kq) it
the state estimate provided by the observer (3.7). This results in the following
control law:
un(kq): Fqi(kq), k_0, 1,2,...
(3.8)
is worth stressing that in order to implement the control (3.8), it is only
necessary to compute the estimate ft(kq) on the kq time scale; in other
It
816
E. W. KAMEN
words, it is only necess ary to operate the obsenrer on the kq time scale.
Hence, it is the gscaled version (3.7) of the obsenrer that we shall utilize in
the study of inputoutput regulation.
Replacing k by kq in the error equation (3.4),
i(kq + q): len
*"
have that
 Lqonl fr(kq), k:
0,
1,2,...,
(3.g)
and inserting (3.8) into the gscaled version of (3.2;, we obtain
: AT*(kq) lrFrft(kq), k
*(kq + q)
But i(kq)

x(kq)
*(kq + q)

i(kq),
: le,
0,
1
,2,....
(3.10)
and thus from (8.10),
TrFrl
"(
kq) + frCrfr(kq),
k
:
0, I
,2,...
.
(3.1 1)
Then combinitrg (g.g) and (3.11), we have that the closedloop system
resulting from the control (3.8) is giver by

*(kq +q)
lrt*,
+q)
:
l
r'"n
k:0,
Aq??,",]l;l il)l
1,2,...
(3.12)
Equation (3.12) is the state model of the closedloop system in the kq time
scale.
Now if the system ( A, B, C) is reachable and strongly obsenzable, then by
Theorems 2.1 and 3.1, for any n X n matrices A and W over R, there exist
matrices Fn and Ln over R such that
I
e"
Lo
lnEn
lnFn
A"  LnQn
f^Frl
wl
( 3.
13)
In other words, the system matrix of the closedloop system (3.t2) on the kn
time scale can be assigned up to the extent indicated by (3.13). In particular,
LINEAR TIMEINVARIANT DISCRETE.TIME SYSTEMS
if a
and W are chosen to
(3.13), we have that
8L7
be zero, then setting q  n in (g.tZ) and using
r): TnFnfr(kn), k0,
ft(kn + n) : 0, k_ 0, 1,2,....
x(kn +
1,2,...,
Combining these two equations gives
x(kn)
k2,3,4,...,
0,
and thus we have deadbeat control on the kn ttme scale. But since
U,(k")
k  0,L,2
F,ft,(kn),
(3. 14)
and i(kn)  x(kn) for k : 2,3,4,..., and since x(k + 1) : Atc(k) +
Bu(k), it must be true that x(k)  0 for k_ 2n,2n + L,2n + 2,... .
Hence, the control (3.14) with observer (9.7) with q  n is a dpadbeat
inputoutput regulator on the k time scale.
Combining the above results with those of the previous section, w€ have
the following result on inputoutput regulation.
Tunonnvr 3.4 . A necessary and sfficient conditi,on fo, the existence of a
dnadbeat inputoutput regulator gioen by (3.8) and (3.7) is that there exi,st an
integer q
Aq
in u;hich
Lo:
co.se,
Po is
a

TrQr
:
PnO
*
the contro, (3.8) u:ith Fo_ Oo and, the obseraer (3.7) u:i,th
ln a'ddition, if the system (A, B,C) i,s
d,eadbeat regulator.
and strongly obseraable, then q canbe set equal to n, Fn canbe set
equal to (ln)'A", oi{ r, can be set equal to A"(On)', where (I;)" is a right
inoerse of Tn and (O,)' is aleft i,ruserse of Qn.
rei,o,6lxsble
4.
APPLICATION TO TRACKINC
In this section, we utilize the results derived in the previous two sections
to solve a $1pe of trachng problem, called setaoi,nt control. We begin with
setpoint control of the state x(k) for the system (A, B,C) defined by (1.1),
818
E. W. KAME}I
(L.2), and then we consider setpoint control of the output y(k), For existing
results on trachtg in the case of systems over a ring of funciionr defined o1 i
parameter set, the reader is referred to the paper by Conte et al,(1992).
Given an nvector
4 over the titrg R, in
state setpoint control the
over R th; forces the state
resPonse x(k) to be equal to x d, the desired set point. Since the state
response of the system is giver by x(k + 1) : Ax(k) + Bu(k), the existence
of a setpoint control u(k) requires that there exists an mvector tJ@ over R
x
objective is to determine a control inpui
u(0
such that
x6:Ax6*Bu,*,
(/  A)*a:
(4.1)
81tr*.
Now, deftning the translations
7(k) : )c(k) 
xd,
n$)u(ft) u*,
we have
*(k + t) :
)c(k
+ t)
 x4:
Ax(k)
* Bu(k) 
xd,
i(k +l):A[ ,(f) +xal+Bla(t)*u*] xd,
x(k+ 1):Ai(/r) +Bn$) + (Ar)*o*Bu*.
Then
if there
exists a u6 over
(4.2)
R satis$ring (+.t), Equation(4.2) reduces to
i(k + 1) 
Afr(k) + Bn(k),
(4.3)
We can write (+.S) in the blockinput form
*(k + q)  Aei(k) + TrUn(k),
(4.4)
819
LINEAR TIMEINVARIANT DISCRETETIME SYSTEMS
where
n$+q
aq$):
1)
u6
a$+q2)
ue
:
a
u(k+
Uq(k)
;*
1)
u@
"(k)
Replacitrg
(4.5)
a
k by kq in (4.4) gives
*(kq + q): Alx(kq) + Tnlr(kq).
nq X n matrix
Now suppose that for some q
Fq
over R. Thet by Theorem 2.3, the state feedbdck control
on(kq)
:  Fq Aq i(kq)
(4.6)
in a deadbeat control on the k time scale. That is, n(k) = Q for
k: Q,cl + L,q + 2,.... Hence, *(k):xa for k: Q,cl + L_,_q +2,...'
results
and so the control (4.6) results in state setpoint tracking. In addition, using
(4.5) we have the the control (4.6) can be rewritten in the form
Ur(kq)
:
 Fq Aq l*(kq)
 * ol + 1"3 uI ' " ull' .
(4.7)
From the above constructions and using Theorems 2,2 and 2.3, we have the
following result.
Turonnrra 4.1. Ci,rsen the desi,red set point x 4 ooer R, there is a deadbeat
setpoi,nt trackl,ng cantrol of the form (4.7) if and qnlA i,f there is an mlsector
l,6.oDerRsuchihot(lA)xa:Bu*,andthereexi,stani'ntegerq>
amntri,x F, over Rsuchthat Aq : loEo. ln addition, i,f the system(A, B,C)
is reachailu, then q can be set uq'uol tu n and Fq can be chosen to be
F*: (I;)'An, where (I;)" is a ri,ght inrserse of I;.
Now given a pvector Ua over R, in output setpoint control we want to
u&) that will force the output y(k) to be equal to U a, the
ftnd a
"onttol
820
E. W. KAMEN
desired set point. The existence of such a control requires that there exist an
mvector n,@ over R and an nvector x@ over R such that
x@:Ax**Bu,*,
Cx*,
Ua:
that is,
:
l';^
f
"lL;:l
;l
A sufficient condition for (4.8) to be satisfied is that
(4.8)
I  A is
R and the p x m matrix c(I  A)rB has a right
inverse
which case a solution to (4.8) is
Lr,@
:
VA
are assuming is the case.
V over R, in
(4.e)
a,
x@: (/  A)'Bu*:
Clearly, right invertibility
invertible over
(I  A)"rAo.
(4.10)
of C(I  /r)LB requires that p 4 m, which we
To see that (4.9), (4.10) is a solution to (+.S), first multiply both sides of
(4.t0) on the left by the matrix I A. This results in the eqlation x@ : Ax*
+ 81tr*.Now multiplyt"g both sides of (4.10) on the left by C gives
Cx*: C(I 
,.1t nVyo: yo.
Hence, the second equation constituting (+.S) is also satisffed.
summarizing the above constructions, we have the following sufficient
conditions for the existence of a deadbeat output setpoint trackei.
THronau
C(l  A)rB
4.2.
Suppose that
I  A is i.noertible
has a right inrserse V ooer
R.
ooer
R, p 4 m, and
A^ko slrppose
thai there is an
: lrF, for some F, ooer R. Then the control
iltteger q
_2 I wch that,As
(+.2) u:lth xo_: (I A)tBu* 'onld u: VAa'results in dead.beat antput

y(k) : Va fo, k : q, q + I,..., and x(k) : x'
ad.dition, if the qstem(e,n,C) ls reachabt"e,thei
q can b9 set equal to n, and Fn can be chosen to be F^ : (l*),A" where (1,),
is a rigft.t intserse of 1,.
setpoint tracki.ng; that is,
fork: q,q + 1,... . ln
we conclude this section by considering the case when the state r(ftq)
cannot be directly measured, and thus must be estimated. In this case we
LINEAR TIMEINVARIANT DISCRETETIME SYSTEMS
BzL
shall utilize the obsenzer givet by (3.7) and the control law (4.7) with x(kq)
replaced by the estimate ft,(kq).
Using the above results and those in the previous sections, we have the
following sufficient conditions for the existence of a deadbeat output setpoint
tracker.
THeonru 4.3. Suppose that I  A is i,noertibl,e ooer R, p 4 m, and
C(l  A)LB has a right inoerse V oaer R. Ako suppose that there is an
i,nteger q > I atch thnt As :lnFr: PnAn for some matrices_F.n 1nd P,
ounl R.'Then the observer (s.z)' *itt Lo: Po and the control (4'7) toith
x&fl: i&fl, xa: (l  a)tBu*, and'u*:'Vga rewlts in d'eadbeat outtriking; that is, y(D:ga for k:2q,2q + !,'., and
put
'x(k)setaoini
: xa for k : 2q,2q + 1,. . . tn a.dli,tion, if the system (L n,C) ts
reachable and strongly obsensable, then q can be set e.qual to n, F, can be set
equal to (l^\'A", ond P* canbe set equal to A"(O,)t, ushere (ln)' is a right
i,nrserse of l^ and (O,)' tis aleft inrterse of O,.
5.
STABILIZATION
A limitation of controltheoretic results based on reachability and strong
observability is that these dynamical properties may be too "strong" for
certain rings R of interest. To achieve more general results, we can consider
schemes that yield a stable closedJoop system rather than attempting to
assign the closedloop dynamics. This of course requires a notion of stability,
whiih can be introduced in a purely algebraic way [e.g., see Khargonekar and
Sontag (1932) and Sontag (1985)1, or can be introduced by adding a topologrcal structure on the .i"g R [e.g., see Byrnes (1980), Kamen and Green
(1980), Green and Kamen (19s5), and Kamen (1985)1.
In this section, we utilize the blockinput form of the state equation to
study the problem of stabilization for systems (e, n,C) deffned over a ring
R, but wh6re now R also has the structure of a normed algebra over the real
R. More precisely, it is assumed that R has a norm ll ll, and R contains a
copy of the reals R, so that R is an Ralgebra. Given an nvector x :
l*, *, "' xnl' ove, R, we deftne the norm of x by
;lrll'
:
llrrll2 + llrnlln +
"'
+llr,ll2.
For any matrix M over R, the norm llM ll is the induced operator norm.
A system (L n,C) over R defined by (1.1), (1.2) is said to be asymptoticaLly stable (AS) if for any initial state r(0) over R
ll
a*"(0)ll
o
as k+."
(5.1)
822
E. W. KAMEN
It is easy to prove (and is well known) that (5.1) is equivalent to the existence
of a positive integ er q such that
llAqll
(5.2)
Suppose that (A, B,C) is AS, and let gmin denote the smallest positive
integer_such that (5.2) holds with q : g^in It is known that
Qmin may be
larger than n, the size of the matrix A. Note that this is in contrast to the
property of reachability in that reachability always implies reachability in n
steps.
Given r!" system ( A, B , C) and a positive integer q
the gscaled blockinput state equati o; (2.2) which is reprod,r"J below:
*(kq + q)
:
AQ
*(kq) + trUr(kq)
(5.3)
.
Then as discussed in Section 2, with the control
un(kq)
 Fq*(kq), k
o, I
,2,...,
(5.4)
the resulting closedloop system in the kq time scale is giver by
*(kq + q) _
l/r,
TrFrl
"(
kq), k
0, I
,2,....
(5.b)
We then have the following concept.
DnnNrrroN
5.1.
The system (A, B,C) is qstep stabilizable for some
integer g
R such that
ileq_qFl
(5.6)
We have the following result on gstep stabilizability.
PnoposrrroN 5.f . Suppose that the system (A, B,C) is q_step stabilizable, .so that there exists an Fq oner R such that (5.6) is satisfi,ua. rhen the
control (5.4) stabilizes the giaen system on the k time scale; that is,
llr(k)ll + 0 for
Proof.
anA
initial state r(0)
orser
R.
Suppose that there exists an Fo over R such that llAq
C
equation (5.5) satisfies the inequality
ll
"(kq)ll < "*ll*(o)ll
, k: L,2,...
 I; F,,ll
823
LINEAR TIMEINIVARIANT DISCRETETIME SYSTEMS
0 as
Then since c
k
+
m, but
it must be shown that
convergence to zero occurs on the k time scale. To see this, first note that
since Ur(kq) : Fqx(kq), llur(tcq)ll + 0 as k + @, which implies that
llz(k)ll * o as k + &. Then since x(k + 1)  Ax(k) + Bu(k), for any initial
state r(0),
ll*(kq+i)Ai*(tilll+0
as
k+oo
for iL,27...1c1
1,
t
llr(k)ll + 0 as k + oo.
By Propositions 5.1, we see that gstep stabilizability is a sufftcient
and thus it must be true that
condition for the existence of a stabilizing state feedback control. As a partial
converse to this result, it turns out that if the system ( A, B, C ) is stabilizable
by memoryless state feedback [i.e., there is an F over R such that x(k + 1)
 ( A  BF)x(k) is AS], then the system is gstep stabilizable for some
q
5.2. If there is an F ooer R such that x(k + 1) : ( A
Bilx!) is AS, then the system (A, B,C) is qstep stabilizable fo, soffLe
PnoposrrroN
q

Proof . Suppose that there exists an F over R such that x(k + 1)  ( A
BF)x(k) is CS. Then there is a positive integer q such that ll(a BF)qll
(A BF)':AQ
lnF,
for some nq X n matrix Fq over R. Thus (5.6) is satisfted, and the proof is
completed.
By Propositions 5.1 and 5.2, the stabilization problem "reduces"
I
to
ftnding an Fq over R such that ll A: TqEqll is minimized and the minimum
value is less'than 1. Of course, there is the problem of selecting q. One
approach would be to start with some value of q such as q  n, and then
r"q,r"tttially increase q until a minimum less than I is achieved. If a
minimum less than I is not found after considering a large range of values of
q, one would expect the "probability" to be high that the system is not
stabilizable.
Of course, if lq has a right inverse (lq)" over R for some q >_L(i."., the
system is reachable), then a minimizing'Fn is F, : (lq)'Aq, and the minimum value is zero. This results in Ceadbeat'control, which was considered in
Section 2.
824
E. W. KAMEN
In the remainder of this
section, we show that the computation
minimiztng Fq can be carried out using a generalized linear quadratic
of a
(LQ)
approach. This construction requires that the normed algebra R have an
inner product given by ( x, A) € R for all x, A c R, and which satisfies the
standard assumptions: For all x, A, z c R and a € R
: (a, x),
(x*z,A) (*,y) + (z,A),
(x, a)
(o*, A)  a(x,
(*,*)As is well known,
if
A)
,
ll"llt.
R is an algebra over the reals R with an inner product
(*,A) that satisfies the additional properties
for
all
x,
e
R
x,
_
and
(*, *)  0 if and only if
0,
then R can be made into a normed algebra by defining the norm by
llrll  ( x, x)tt'.
In addition to the inner product operation, we also require a *operation
from R into R with the properby that
(x,
Az)
 (y* x, z)
for
all
x, U, z € R.
Examples of normed algebras with a *op"ration that arise in applications are
given in Green and Kamen (tgAS) and Kamen (lg85).
Finally, the inner product, norm, and *op"ration can be extended to
vectors and matrices over R as follows. Given the nvectors x
l*, xz "' *nf' and A  lA, Uz ... Unf',we deftne
n
\l
/r (*n,
(*, a)
llrll
i:I
:
(x,
u),
*)t/'
LINEAR TIME.INVARIANT DISCRETETIME
SYSTEMS
 (rprj) over R, we deftne lryx
P
overR,
anynXn matrix
and given a matrixW
:
(P*, A)
(x,
P*

825
(u:fi).Note that for
A).
(5.7)
Now grven a mq X mq u:eightingmatdxW over R, consider the quadratic
cost functional
I* 
ll
"
k_0, 1r2r...,
(kq + q) ll' +llwu,(kq) ll'
(5.8)
where x(kq) is given by (5.3). The objective is to solve the generalized LQ
problem; that is, we want to compute a block input {.lr(kq) over R that
minim rzes the cost l * for each k
( A, B , C ) is given
Tseonnv
in the next result.
5.1. lf for son'Le integer q
1ryxW is imsertible ooer R, then a control Ur(kq) that mini,mi,zes J7,
un(kq)
:
 [ ri rn + 1ry *w ]'ry^ ' *(kq)
Proof. In terms of the inner product (','), the
cost
i,s
(5.e)

/1 can be written in
the form
(*(kq + q), x(kq + q)) + (wur(kq),wur(t
I*:
Inserting the expression (5.3) for x(kq
+ q) into (5.10) and
q)).
(5.10)
using (5.7) gives
Ir,: ( "( kq) ,( Ar)* A, *(kq)) + z(*(kq), ( A')* trUr(t q))
+ (ur(kq) , [s fn + 1ryxw]un(kq)).
Then
gives
if ryq + W*W
I^:
ll
tn
$il
*
(
is invertible over R, "completing the square" in
+ [rirn + zryxw]'t;^ n*(t
" $q),
(
A,)*le,
rn(S rn +
illl'
1ryxw
)' t;^ rf*f tq))
]r
826
It
E. W. KAMEN
is then obvious that the minim tzingcontrol is given by
It is interesting to note that the optimal
(5.9).
t
control (5.9) is in the form (5.4)
where
Fq: tryq + w*wl*'r;e,
(5.1 1)
Also, one should note the simple form of the feedback gain Fn; in particular,
Fn
be com,puted without having to solve a Riccati equatioh, as one must
2an
in the standard LQ approach to systems over the reals. As seen from the following result, with a proper selection of the weighting
w, the control (5.9) is a stabilizing control if the system is gstep stabili"zablel
5.2. 9rppose that the system (A, S,C) is qstep stabilizable
q 2 r. Let w : or/zln*, where o is a strictlE positi.ae real
for
number' Then i,f o is sufficiently small'and $lo + clo^ is {noertible orser
Tsnonnu
some integer
R, the control (5.9) results in a closedloop tyitnh that & eS.
Proof.
.S,1ppose
over R such that
control is
ll
1,,:
that there is a control Ur(kg)^: Gqx(kq) with
,qq lqcqll
ll(
^n
rrGn)
"(
t
illl' + ollcr*(kq)il'
Gq
(5.12)
Let Fo denote the feedback gain of the optimal control with W : aL/zlo*.
Then since the cost with the optimal control cannot be greater than the cost
(5.L2), it must be true that
ll(
rnFr)
^'
"(
rrq)ll'
+ olltn*(kq)
ll' *
ll(
a'
r,G,)
"(
rrilll'
+ ollc,*(kq) ll'
Thus, since
lleq
ll
A,
Tqcqll
lnErll < 1. Therefore, by Proposition S.r the resulting closedloop
AS.
I
:_o)"r*
With W
and the feedback gain Fq given by (5.11), the system
matrix Aq TnFn of the resulting closedloop system on the kq time scale is
system is
given by
Aq TrFr: ft
tn(ryq + orr*)'ry]r^,
(5.13)
LINEAR TIME.INVARIAN{T DISCRETETIME SYSTEMS
827
Here we are assuming that the inverse ir, (5.13) does exist. Now by Theorem
5.2, we know that the closedloop system with the system matrix given by
(5.13) is AS if the system is gstep stabilizable and a is chosen to be
sufftciently small. Hence, it is possible to test for stabilizability (modulo the
invertibility requirement) by chechng to see if the norm of
Ir, L(ryq + crq*) r r;7 o,
is less than
I
for some integer value of q and some positive value of o.
6. CONCLUDINC COMMENTS
It
is clear that the results in Section 5 can be dualized to yield correspondresults on the construction of blockform observers deftned by (3.3) *ittt
error dynamics that are AS. In addition, the stabilization results can also be
applied to the design of inputoutput regulators grven by the observerstatefeedbackcontroller combination &scussed in Sections 3 and 4. Via this
approach, new results on regulation and setpoint tracking can be obtained
without having to require reachability and strong observability of the glven
system. The details are not pursued here.
itg
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Sontag,
Recehsed B
May J993; final marwscri,pt accepted 20 September 7993
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