DTIC ADA439986: Stabilization of LTI Systems with Communication Constraints

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ECHNICAL ESEARCH EPORT
Stabilization of LTI Systems with Communication Constraints
by Dimitrios Hristu
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Stabilization of LTI Systems with Communication
Constraints1
Dimitris Hristu2
University of Maryland,
Institute for Systems Research,
College Park, MD 20742
e-mail: [e-mail]
tational complexity and are appealing because they
Abstract
quantify the amount of \attention" that the controller
paystoeachcomponentoftheunderlyingsystem. Pre-
This workis directed towardsexploring interactionsof
vious work has addressed problems involving stabiliz-
communication and control in systems with communi-
ing anLTIsystemwhenonlysomeofits output(s)can
cation constraints. Examples of such systems include
be measured at one time [10]. That work explored the
groupsofautonomousvehicles,MEMSarraysandsys-
e(cid:11)ects of communicating sequentially with each of a
temswhosesensorsandactuatorsaredistributedacross
set of linear systems and described a method for sta-
anetwork. Weextendsomerecentresultsinvolvingsta-
bilizing all systems in the set. This paper extends and
bilizationofLTIsystemsunderlimitedcommunication
completes those results, taking into account communi-
and address a class of feed-forward control problems
cation constraints that are present both when making
for the systems of interest.
measurementsaswellaswhentransmitting controlac-
tions to the LTI system.
1 Introduction
We present a new \extensi(cid:12)cation algorithm" that
With communication networks proliferating into in- transforms the stabilization problem to an equivalent
creasinglymanyaspectsofmodern-daytechnology,en- problem involving matrix search (as is the case with
gineers are faced with new opportunities involving the the work in [10]). Our goal is to combine our algo-
control of systems whose components are distributed rithm together with recent results on tracking for net-
acrossa networkandwhicharesupposedto operatein worked control systems ([8]) so that closed-loop track-
a coordinated manner. Examples of such systems in- ing can be achieved for the systems of interest. For
clude groupsof vehicles, satellite clusters,mobile com- related problems in hybrid system theory see [3], [5]
munications,MEMSarraysandothers. Thechallenges [4], and [12]. Work discussing systems with communi-
that these systems presentgo beyondproblems associ- cation constraints can be found in [2], [11] and others.
ated with increased dimensionality. Frequently, their
performance is limited not because of lack of compu-
2 A Prototype Computer-Controlled System
tational power but because of lack of time on a shared
network of sensors and actuators. This has lead to
Consider an n-dimensional LTI system G(s) with
recente(cid:11)ortstowardsbringingtogetheraspectsofcon-
input u 2 m, and output y 2 p. The system
trol and communication under a framework that will
R R
is driven by a computer or other digital controller
lead to a better understanding of controlsystems with
(Fig. 1) that may be remotely located and which does
communication constraints.
not have simultaneous access to all inputs/outputs of
One way to jointly formulate control and communica- the control system. In particular:
tion problems is to employ the idea of a \communica- (cid:15) The controller sends commands to and receives
tion sequence" [6] which allows multiple (sub)systems inputs from the system every (cid:1) units of time, via a
to share the attention of a centralized controller [8]. zero-order-holdstage.
Communicationsequencesbringforthsomeofthecon- (cid:15) Inputs/outputs are communicated through a bus
nections between control, communication and compu- which has limited capacity. Speci(cid:12)cally, the bus can
\carry"at mostb>0 signals, with b<m+p, b2 (cid:3).
1ThisworkwasfundedbyNSFGrantno. EEC94-02384and N
AROGrantno.DAAG55-97-1-0144. Typically, the capacity of the communication bus
2This work was done while the author was with the Divi-
will be divided between input and output signals,
sionofEngineeringandAppliedSciencesatHarvardUniversity,
Cambridge,MA. with br channels devoted to sampling the output of
and with the transmission of those samples through
the communication bus.
De(cid:12)nition 2 Consider a computer-controlled system
G(z) with b < b (b < b) being the dimension of the
w r
input (output) communication bus. A pair of commu-
nication sequences σ 2 m(cid:2)N, σ 2 p(cid:2)N is admis-
w r
E E
sible if:
Figure 1: A closed-loop computer-controlled system (cid:15) Pkσw(i)k2 (cid:20)bw,Pkσr(i)k2 (cid:20)br 8i=0,...,N −1
(cid:15) m σ (i)+ p σ (i)(cid:20)b 8i=0,...,N −1
j=1 w j=1 r
(cid:15) Spanfσ (0),...,σ (N −1)g= m and
the underlying LTI system and bw channels used for Spanfσw(0),...,σ w(N −1)g= Rp
transmitting control inputs. We will refer to these r r R
Theaboveconditionsrequirethatnomorethanb (b )
two groups of channels as the \input" and \output w r
ofthesysteminputs(outputs)areupdated(measured)
bus." Of course, b and b may change at any time as
r w
by the controller at every step and that the pair
long as b +b = b. This represents a rather general
r w
(σ ,σ ) allows communication with all inputs (out-
setting for controller/plant communication, allowing w r
puts) of the linear system at least once every period.
for dynamic recon(cid:12)guration of the available channels.
We will use the terms \narrow" (b < m or b < p)
w r 3 Stabilization with Limited Communication
and \wide" (b (cid:21) m and b (cid:21) p) to describe the
w r
communication bus.
We now focus on the problem of stabilizing systems
Inthefollowingitwillbeconvenienttousethediscrete- like the one discussed in the previous section, using
time equivalent of the linear system G(s): static output feedback (see Fig. 1). We will take the
numberofinputandoutputchannels(b andb )tobe
w r
x(k+1) = Ax(k)+Bu(k) constant.
y(k) = Cx(k) (1)
Problem Statement 1 Given: a computer-
Each element of u retains its value (by virtue of the controlled system G(z) with b ,b 2 N(cid:3),b +b = b
ZOH stage) until that element is updated by the con- r w r w
denoting the size of the input and output commu-
troller. At the same time, the controller may only re-
nication busses and a pair of admissible N-periodic
ceivepartialinformationabouttheoutputy(k). When communication sequences σ ,σ 2 p(cid:2)N, m(cid:2)N, (cid:12)nd
the communication bus is narrow, one possibility is to a constant feedback gain Γ 2r wp(cid:2)mEtpheart sEtapbeirlizes the
choose a sequence of operations for the switches (see
R
system under output feedback.
Fig. 1) that select which inputs/outputs are to be up-
dated/sampled at a particular time. This leads us to
the notion of a \communication sequence" (originally In [10], weshowedthat a versionofthis problem(with
introducedin[6])bywhichthecontrollerchooseswhich noconstraintsontransmittingcontrolsamplesu(k))is
of the input signals (measurements) to update (read) equivalent to the NP-hard problem:
at every step.
Problem Statement 2 Given a collection of matri-
ces A 2 q(cid:2)q,0(cid:20)i(cid:20)i , and scalars γ ,...,γ 2
De(cid:12)nition 1 An N-periodic communication se- i R max 1 imax
, (cid:12)nd a stable element of the a(cid:14)ne subspace
quence is an element of
R
Epme(cid:2)rN =f(σ(0),σ(1),...,σ(N −1), A=A0+iXmaxγiAi (3)
σ(0),...,σ(N −1),...):σ(i)2f0,1gmg (2) i=1
whereq =(2N2−N)nandi =mp. Inthenextsec-
max
for some m>0.
tion we present a new extensi(cid:12)cation algorithm. This
We will consider the controller’s communication with algorithm is complete in the sense that takes into ac-
the system to follow a periodic pattern speci(cid:12)ed by count the both input (b <m) and measurement con-
w
a pair of N-periodic sequences: a \control" sequence straints (b < p) and arrives at a similar construction
σw 2 pme(cid:2)rN will be used to transmit inputs and a for the martrices that span the a(cid:14)ne subspace of inter-
\measuErement" sequence σ 2 p(cid:2)N will provide a est.
r per
E
patternforsamplingthesystemoutput. Theentriesof
σ (i)(σ (i))indicatewhichelements ofu(k)(y(k))are 4 Extensive Form of a Discrete LTI System
w r
to be updated (measured) at the kth time step. We
willignorequantizationerrorsassociatedwiththerep- Consider the system of Fig. 1, to which a static out-
resentation of signal samples in the digital controller put feedback controller is attached, subject to limited
p. 2
communicationasdescribedinSec.2. Iftheinput and 4.2 Constrained Control
output busses were both \wide" (b = p, b = m) We now follow the procedure of Sec. 4.1, applied this
r w
thenthestaticoutputfeedbackcontrolu(k)=ΓCx(k) time to communication over the input bus. The in-
wouldbepossibletoimplementandΓwouldhavetobe put vectors u(k) arriveat the LTI system accordingto
chosen to make the closed-loop dynamics (A+BΓC) the communication sequence σ 2 m(cid:2)N. At the kth
w per
stable. We proceed to modify this simple situation to step, only the inputs speci(cid:12)ed by thEe non-zero entries
reflect the existence of communication constraints. of σ (k) are updated, with all other inputs remaining
w
at their previous values:
4.1 Constrained Measurements
Assume for now that bw = m so that the controller u(k)=diag(σw(k))Γyl(k)+(I−diag(σw(k)))u(k−1)
cantransmittheentireinputvectoratoncetotheLTI (9)
system. Because measurements are to be obtained ac- By iterating backwardsfor a full period (N steps) and
cording to the communication sequence σr 2 ppe(cid:2)rN, assuming that the communication sequence σw is ad-
only some of the elements of y(k) are received bEy the missible (i.e. all m elements of the input are updated
controller at each step k. The elements for which the at least once every N steps) we obtain:
controllerdoesnotreceiveupdatesareassumedtohold
NX−1
theirlast-knownvalues. Moreprecisely,wehaveacon- u(k)= D (k,i)Γy (k−i) (10)
W l
trol law of the form
i=0
u(k)=Γyl(k) (4) where
(cid:26)
whereyl(k) is the output vector composedof the most 4 diag(σw(k)) Q i=0
up to date information available to the controller at DW(k,i)= diag(σ (k−i)) i−1M (k,j) i>0
the kth step. Notice that in general yl(k) 6= y(k) be- w j=0 W (11)
causenotallelementsofy(k)arecommunicatedtothe 4
arediagonalm(cid:2)mbinarymatricesandM (k,j)=I−
controller at step k. We can write W
diag(σ (k−j)). ThejthdiagonalelementofD (k,i)
w W
y (k)=diag(σ (k))y(k)+(I −diag(σ (k)))y (k−1) is 1 if the jth input was last updated at the (k−i)th
l r r l
(5) step, 0 otherwise.
where for a vector x2 n, diag(x) is an n(cid:2)n matrix
with the elements of xRalong its diagonal and all o(cid:11)- 4.3 Combining Communication Constraints
diagonal entries being zero. By iteratively applying If communication with the system is to proceed ac-
Eq.5foranumberofstepsequaltothecommunication cording to the pair (σr,σw) of admissible N-periodic
period N, we obtain sequences, we can combine the results of Sec. 4.1 and
Sec. 4.2. By substituting Eq. 7 into Eq. 10 we obtain:
y (k) = diag(σ (k))y(k)+
l r 0 1
2XN−2
NX−1diag(σ (k−i))@iY−1M (k,j)Ay(k−i) (6) Bu(k) = Fkix(k−i) (12)
r R
i=0
i=1 j=0
where
4
where M (k,j) = I − diag(σ (k−j)). We observe
R r bic(Xi−N−1)
that if the communication sequence σr is admissible F =4 B N D (k,j)ΓD (k−j,i−j)C (13)
then each of the p elements of the output y(k) will be ki W R
read at least once every N steps so the summation in j=min(i,N−1)
Eq.6terminatesafteratmostN steps: systemoutputs We can now write the closed-loop dynamics of the
older than N steps areessentially\overwritten"in the computer-controlled system as:
controller. We can rewrite Eq. 6 as
2XN−2
NX−1 x(k+1)=Ax(k)+ F x(k−i) (14)
y (k)= D (k,i)Cx(k−i) (7) ki
l R i=0
i=0
Letus de(cid:12)ne Comp(p)to bethecompanionfoPrmasso-
where ciated with an nth-degree polynomial p(s)= np si
(cid:26) 0 i
4 diag(σr(k)) Q i=0 2 3
DR(k,i)= diag(σ (k−i)) i−1M (k,j) i>0 0 1 0 (cid:1)(cid:1)(cid:1) 0
r j=0 R 66 0 0 1 (cid:1)(cid:1)(cid:1) 0 77
are diagonal p(cid:2)p matrices with binary entries. T(h8e) Comp(p)=4 666 ... ... ... ... 0 777 (15)
jth diagonal element of DR(k,i) is 1 if the jth output 4 0 (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) 0 1 5
was last read at the (k−i)th step and is 0 otherwise. pn pn−1 (cid:1)(cid:1)(cid:1) p1 p0
p. 3
If we now use the F (Eq. 13) to de(cid:12)ne the matrix of the state. If all elements of u(k) could be commu-
kj
polynomial nicated simultaneously (wide input bus), it would be
su(cid:14)cient to \extensify" the state by considering only
2XN−2 its N-past values at times k,k−1,...,k−N −1 (see
f (s)=A+ F si (16)
k kj [10]). However, because the inputs are also subject to
i=0 delays in arriving at the LTI system, a particular el-
then the closed-loop dynamics of Eq. 14 can be ex- ement of u(k) could persist for (at most) the next N
pressed in (cid:12)rst-order form: steps k,k+1,...,k+N −1. This implies that the en-
tries of u(k) as seen by the LTI system could depend
χ(k+1)=Comp(f )χ(k) (17) on state values that are (at most) 2N −1 steps old.
k
where χ=[xT (cid:1)(cid:1)(cid:1) xT xT ]T 2 (2N−1)n. In summary, we have given a procedure for converting
(k−2N+1) (k) (k+1) R anoutputfeedbackstabilizationprobleminvolvingLTI
The system of Eq. 17 is linear, time-varying and de- systems under limited communication, into a search
scribes the state evolution of the computer-controlled problem involving a (cid:12)nite collection of (2N2 − N)n-
system under output feedback and periodic commu- dimensional matrices. These matrices are obtained
nication. We have essentially \extensi(cid:12)ed" the state from the parameters of the control system together
vector to include past values up to two communica- with a pair of admissible communication sequences.
tion periods. We note that the matrix Comp(f ) is
k
N-periodic in k, so Eq. 17 represents a periodic linear Equations 18 and 19 show that after a full communi-
systemof dimension (2N−1)n. Although oflargerdi- cation period, the stability properties of the extensive
mension,this periodicsystemis equivalentto the orig- form can be captured in a lower-dimensional space by
inal system in the sense that it exactly describes the considering
closed loop dynamics under the communication policy z(k+1)=A^z(k) (21)
that was imposed.
whereA^=AN−1AN−2...A0anddim(z)=(2N−1)n.
In this lower-dimensionalspace,thenumber ofgainsis
It is a fact that every discrete-time periodic system
the same as in the extensive form, however the gains
can be expressed as a time-invariant system of higher
dimension(see[7]). Inourcase,this yields a systemof enter A^nonlinearly. Ifthenumber ofgainsisnγ =4 mp
order (2N2−N)n which we call the \extensive form" then the number of matrix coe(cid:14)cients for the exten-
of the original system in Problem 1: siveform is nγ+1. In the lower-dimensPional(cid:0)spaceth(cid:1)e
numberofmatrixcoe(cid:14)cientswouldbe N nγ+i−1 .
Xe(k+1)=AXe(k) (18) Thus, even though lower-dimensional mati=ri0ces arie of-
ten convenient from a computational viewpoint, the
where Xe(k)2 (2N2−N)n and potential savings in algorithm run times are o(cid:11)set by
R
2 3 the memory demands due to the large number of coef-
6 Com0p(f1) (cid:1)0(cid:1)(cid:1) (cid:1)0(cid:1)(cid:1) 00 Com0p(f0) 7 (cid:12)cient matrices that Eq. 21 requires.
A=4 0 Comp(f2) 0 (cid:1)(cid:1)(cid:1) 0 5
... ... ... (cid:1)(cid:1)(cid:1) 0 5 An Extensi(cid:12)cation Example
0 (cid:1)(cid:1)(cid:1) 0 Comp(fN−1) 0
(19)
Consider the scalar system
By construction, stability of the extensi(cid:12)ed system
(Eq. 18) is equivalent to the stability of the original x(k+1) = ax(k)+u(k) (22)
system. Moreover, each of the matrices Comp(f ) are
a(cid:14)neintheentriesofΓ.PBychoosingabasisfor km(cid:2)p, y(k) = x(k) (23)
wecanexpressΓasΓ= mp γ E whereE isanRm(cid:2)p
i=0 i i i We assume that the controller communicates with the
matrixwhose(bic+1,imodp+1)th entryis\1",with
p abovesystemoverabusofwidthb=1accordingtothe
allotherentriesbeingzero. InthebasisofthefEigwe following pair of 2-periodic communication sequences:
can express A as an element of the a(cid:14)ne subspace
σ =(1,0,1,0,...), σ =(0,1,0,1,...) (24)
Xmp r w
A=A + γ A (20)
0 i i Noticethattheonlycommunicationchannelisusedfor
i=0
measurementsoftheoutputandfortransmissionofthe
where each of the A are obtained by substituting E control actions in an alternating fashion. Clearly, the
i i
for Γ in Eq. 13. above sequences are admissible. We want to stabilize
the system using a control law of the form
We note that at the kth step, the latest available out-
puttothecontrollery (k)dependsonthepastN values u(k)=γy(k−d(σ ,σ ,k)) (25)
l r w
p. 4
where d(σ ,σ ,k) is a delay that depends on the com- 7 Simulation Results
r w
munication sequences and the current step k.
Consider the fourth-order LTI system:
The scalar system we began with, combined with the (cid:20) (cid:21) (cid:20) (cid:21)
1 3/4 1/2 0 0 0
period-2 pair of communication sequences σr,σw gives x(k+1) = 1/4 3 1/3 −1/3 x(k)+ 1 0 u
1/6 0 −1/2 −3/7 0 0
rise to a 3-dimensional periodic system χ(k + 1) = (cid:2) 0 −1 2/(cid:3)5 0 0 1
Comp(f )χ(k) with y = 1 0 0 1 x (31)
k (cid:26) 0 1 0 0
a+γs2 k even whose open-loop system eigenvalues are shown in
f (s) = (26)
k a+γs k odd Fig. 2. We want to stabilize this LTI system using
The corresponding companion forms are:
3
8 2 3
>>>>>>< 4 00 10 01 5 k even 2
1
2 γ 0 a 3
Comp(f ) = (27)
k >>>>>>: 4 00 10 01 5 k odd Im 0
0 γ a −1
−2
and the extensive form is given by the 6-dimensional
LTI system: −3
2 3 −2 0 Re 2 4
0 1 0
6 7
6 0 0 0 1 7 Figure2:Eigenvaluesofopen-loopsystem,kλ k=3.2.
6 7 max
Xe(k+1)=666 0 1 0 γ 0 a 777Xe(k) (28) static output feedback, given that the communication
4 5
0 0 1 0 bus can carry two signals to/from any of the inputs or
0 γ a outputs (i.e. b = 2). Of the two available channels,
one is to be used for transmitting control values, the
Choosing the gain γ in order to stabilize the above
otherforobtainingmeasurements. Inthefollowing,we
system is equivalent to (cid:12)nding a stable element of the
investigate the performance of the extensi(cid:12)cation and
formgivenby Eq.20 whereA andA canbe reado(cid:11)
0 1 simulated annealingalgorithms(Sec. 4,6 ) for two dif-
from Eq. 28.
ferent communication sequences.
6 Finding a Set of Stabilizing Gains
7.1 Control with Uniform Attention
Initially thecontrolleris to divideits attentionequally
Afteracomputer-controlledsystemhasbeenputinthe
between each of the input/output pairs (u ,y ) and
1 1
extensiveformasdescribedinSec.4,theoutputstabi-
(u ,y ). For that purpose, we chose
2 2
lizationproblembecomesequivalentto(cid:12)ndingastable (cid:18)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:19)
element of the a(cid:14)ne subspace de(cid:12)ned by Eq. 20. This σ =σ = 1 , 0 , 1 , 0 ,(cid:1)(cid:1)(cid:1) (32)
isanNP-hardproblem[1]. Onepossibilityis toPchoose w r 0 1 0 1
thegainsγ sothattheeigenvaluesofA=A + γ A
i 0 i i i to be thecommunicationsequencesto be used,so that
are enclosed in a circle with the smallest possible ra-
the controller reads and transmits in an alternating
dius. This suggests minimizing the spectral radius of
pattern.
the closed-loop system
The matrices composing the extensive form were com-
η =jjλ (A)jj (29)
max puted and simulated annealing was performed on the
To negotiate the large number of local minima that four elements ofthe feedbackmatrix Γ. Thesimulated
are expected, we use simulated annealing on the gains annealingalgorithmwasstoppedafter5000steps. The
γ . Our algorithm numerically computes the gradient coolingschedulewasa 1 decayfollowedbyalinear
i log(n)
∂η/∂γi and then lets the gains γi flow along that gra- decaytozerowhenthegaing(t)reachedapre-speci(cid:12)ed
dient, adding a white-noise term dw with a gain g(t) level of 0.1. A plot of the cooling schedule is shown
that decays to zero: in Fig. 3. Choosing \good" cooling schedules for the
stabilization problem considered here remains an open
∂η
dγi = dt+g(t)dw. (30) problem. Inthiscase,thespectralradiusdidnotreach
∂γ
i valuesbelowunity. Theevolutionofthespectralradius
The\coolingschedule"g(t)shouldgotozeroast!1, isshowninFig.4. Theresulting(cid:12)naleigenvaluesofthe
butitshoulddosoataslowenoughrateforthespectral extensi(cid:12)ed system are shown in Fig. 5, corresponding
radius to approach the global minimum. to a closed-loop system that was unstable.
p. 5
7.2 Towards Optimal Communication
Next, we investigated a period-four pair of communi-
cation sequences,
Cooling Schedule
50
(cid:18)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:19)
1 1 1 0
40 σw =σr = 0 , 0 , 0 , 1 ,(cid:1)(cid:1)(cid:1) (33)
30
that devote three cycles to the pair (u ,y ) for every
1 1
20
one cycle allocated to (u ,y ). The above sequences
2 2
were chosen after some experimentation and by notic-
10
ing that the upper-left 2(cid:2)2 block of the dynamics for
thestateevolution(Eq.31)hasalargerspectralradius
0
0 1000 2000 3000 4000 5000
than thelower-rightblock (whenthe couplingbetween
the two blocks is removed). As a result, communicat-
Figure 3: Cooling schedule.
ingmoreoftenwiththe(u ,y )pairmayleadtobetter
1 1
performance.
Thecoolingschedulewasthesameasintheuniformat-
tention case. This time, simulatedannealingstabilized
the closed-loopsystem, reducing the spectralradius to
0.795. The (cid:12)nal closed-loop eigenvalues are shown in
Spectral Radius
Fig. 6. with the evolution of the spectral radius of the
6
5 1
4
0.5
3
2
m 0
I
1
00 1000 2000 3000 4000 5000 −0.5
Figure 4: Evolution of spectral radius (uniform atten-
−1
tion). −1 −0.5 0 0.5 1
Re
Figure 6: Closed-loop eigenvalues with non-uniform at-
tention, λ =0.795.
max
system shown in Fig. 7.
3
Spectral Radius
2
4
1 3.5
m 0 3
I
2.5
−1
2
−2 1.5
1
−3
−4 −2 0 2 4 0.5
Re
0
0 1000 2000 3000 4000 5000
Figure5:Closed-loopeigenvalueswithuniformattention,
kλmaxk=2.7. Figure 7: Evolution of spectral radius (non-uniform at-
tention).
p. 6
8 Combining Stabilization with Feed-forward where
Control 2 3 2 3
ψ(1) v(0)
6 7 6 7
We outline a method for combining stabilization and ψ~=4 66 ψ(.2) 77 v~=66 v(.1) 77, (40)
4 . 5 4 . 5
feed-forward control. Consider an LTI system subject . .
to communication constraints as described in Sec. 2 ψ(µ) v(µ−1)
and assume that a stabilizing feedback matrix Γ has
been computed. Let fydgkkm=1, (yd(k) 2 Rp,km > 0 be µ=km/(2N −1) and
a desired output sequence of (cid:12)nite duration, which we
4
want to track with the output of the system of Fig. 1. (cid:3)=[(cid:3) ] (41)
ij
That is, we allow the controller to send signals of the
with each (2N −1)p(cid:2)(2N −1)m block (cid:3) given by
form ij
u(k)=Γy(k−d(σ ,σ ,k))+u0(k) (34) 8 (cid:16)Q (cid:17)
w r >< C (i) j A (q) B (j) i>j
where d() is a delay due to the communication con- c q=i c c
straints. The signal u0(k) is a feed-forward term (cid:3)ij =>: Cc(i)Bc(j) i=j (42)
0 j >i
whichissubjecttothesamecommunicationconstraints
(σr,σ(w)) as the feedback signal so that it may be su- If p (cid:21) m, (cid:3) has more rows than columns. If (cid:3) has
perimposed on the stabilizing inputs transmitted by
maximal column-rank, then we can use its generalized
the controller as Eq. 34 shows. We want to select the
inverse to compute the feed-forward control sequence
sequence u0(k),k =0,...,k so that the error:
m that will minimize the tracking error (Eq. 35) by:
Xkm v(cid:3) =((cid:3)T(cid:3))−1(cid:3)T(ψ −ψ ) (43)
jjy (k)−y(k)jj (35) d ic
d
i=0 whereψ isthee(cid:11)ectofinitialconditionsfortheplant
ic
state x(0). The vector ψ will be comprised of the
is minimized. In the following, we assume that the ic
samples of the sequence
duration (k ) of the desired signal is a multiple of
m
(2N −1).
y (k)=CAkx(0) (44)
ic
If the desired output y (k) is known a-priori, then
d according to Eq. 37, 40.
we can make use of operator-based approaches [8] for
this \preview tracking" problem, suitably modi(cid:12)ed for
Therearetwonecessaryconditionsfor(cid:3)tobeinjective.
discrete-time systems. In short, consider the LTI sys-
First, the original system (Eq. 1) have normal rank:
tem discussed in Sec. 2, in its periodic form of Eq. 17
limsupjzj=1rank(C(zI −A)−1B)=m (45)
χ(k+1)=A (k)χ(k) (36)
c
Second, the communication sequences σ ,σ must be
r w
with Ac(k) = Comp(fk) N-periodic, de(cid:12)ned by admissible,topreventfromconsistently\ignoring"any
Eq. 15, 16. The dynamics of the stabilized system oftheinputsoroutputs. Althoughtheseareonlyneces-
(Eq. 36) must be modi(cid:12)ed to reflect the presence of saryconditions,isispossibletomodifymatterssothat
the feed-forward signal u0(k). Let they are also su(cid:14)cient, by rede(cid:12)ning v~(k) in Eq. 39 to
be: 2 3
v(k) =4 [uT (cid:1)(cid:1)(cid:1) uT uT ]T v(−2N −1)
(k−2N+2) (k−1) (k) 6 . 7
ψ(k) =4 [yT (cid:1)(cid:1)(cid:1) yT yT ]T (37) 66 .. 77
(k−2N+2) (k−1) (k) 66 v(−1) 77
6 7
then v~=6 v(0) 7 (46)
6 7
6 v(1) 7
6 7
χ(k+1) = Ac(k)χ(k)+Bc(k)v(k) 4 .. 5
.
ψ(k) = Cc(k)χ(k) (38) v(µ−1)
where B 2 (2N−1)n(cid:2)m and C 2 (2N−1)p(cid:2)(2N−1)n and (cid:3) to be the new map that takes v~ to y~. This
c c
will be determRined by the choice of cRommunication se- corresponds to considering the \past" 2N −1 samples
quence andthe parametersof the underlying liner sys- of v in order to ensure injectivity of the new map (cid:3).
tem. From a least-squarespoint of view, if the costfunction
penalizes the deviation from a given output y(0), then
Now,feed-forwardinputsequencesaremappedto out- the controller must be given the chance to a(cid:11)ect y(0).
puts by Thiscanonlybeguaranteedifweareallowedtochoose
ψ~=(cid:3)v~ (39) the 2N −1 inputs preceding that output. This is not
p. 7
surprising if we recall the n-step controllability results [2] V.S.BorkarandS.K.Mitter. LQGcontrolwith
for discrete-time LTI systems. If the above conditions communicationconstraints. TechnicalReportLIDS-P-
aresatis(cid:12)edthennon-zeroinputssentbythecontroller 2326, MIT, 1995.
to the LTI system will produce non-zero outputs and
[3] M. S. Branicky,V. S. Borkar,, and S.K. Mitter.
(cid:3) will be kernel-free. We will not give a formal proof
Auni(cid:12)edframeworkforhybridcontrol: modelandop-
of this here; a moredetailed argumentcan be found in
timalcontroltheory. IEEE Transactions on Automatic
[9].
Control, 43(1):31{45,January 1998.
We expect to include results demonstrating combined [4] R. W. Brockett. On the computer control of
tracking and stabilization tasks in the (cid:12)nal version of movement. In Proceedings of the 1988 IEEE Confer-
this manuscript. enceonRoboticsandAutomation,pages534{540,April
1988.
[5] R. W. Brockett. Hybrid models for motion con-
trol systems. In H. Trentelman and J.C. Willems, edi-
9 Conclusions and Future Work
tors, Perspectives in Control, pages 29{54.Birkh¨auser,
Boston, 1993.
In this paper we have addressed the stabilization of
LTI systems which are operating under limited com- [6] R.W.Brockett. Stabilizationofmotornetworks.
munication. Our approachis based on the use of peri- In Proc. of the 34th IEEE Conference on Decision and
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controlandmeasurementsignalsfromthecontrollerto [7] Jon H. Davis. Stability conditions derived from
theplantacrossanetworkorcommunicationbus. The spectral theory: discrete systems with periodic feed-
work presented here builds on previous versions of the back. SIAM Journal of Control, 10(1):1{13, February
extensi(cid:12)cationalgorithminordertohandleconstraints 1972.
a(cid:11)ecting both control and measurements. There are
[8] D. Hristu. Generalizedinversesfor (cid:12)nite-horizon
several issues that present opportunities for further
tracking.ToappearinProc.ofthe38IEEEConference
workrelatedtotheextensi(cid:12)cationalgorithm,including
on Decision and Control, December 1999.
methods for (cid:12)nding \good" communication sequences
and cooling schedules. In addition, it would be desir- [9] D. Hristu. Optimal Control with Limited Com-
abletoinvestigateboundsforthespectralradiusofthe munication. PhD thesis, Harvard University, Div. of
extensive form so that stopping criteria for our simu- Engineering and Applied Sciences, 1999.
lated annealing algorithm can be constructed. [10] D. Hristu and K.Morgansen. Limited communi-
cationcontrol.SystemsandControlLetters,37(4):193{
We have outlined a method for combining basic feed-
205, July 1999.
forward control with the extensive form. Closed-
[11] P. Voulgaris. Control of asynchronous sampled
loop tracking is a logical next step that remains to
data systems. IEEE Transactions on Automatic Con-
be addressed. Finally, although our model for lim-
trol, 39(7):1451{1455,July 1994.
ited communication assumes timely arrival of each in-
put/outputsamplefrom/tothecontroller,thismaynot [12] W.S.WongandR.W.Brockett.Systemswith(cid:12)-
bethecaseinmayrealisticsituations(i.e. controlover nitebandwidthconstraints-partII:Stabilizationwith
the Internet). Current e(cid:11)orts are focused on develop- limited information feedback. IEEE Transactions on
ing models that can handle variability in the arrivals Automatic Control, 42(5):1049{1052,May 1999.
ofcontrolsamplesaswellas\lost"samples thatfailto
arrive at their destination.
10 Acknowledgment
The author would like to thank Prof. R. W. Brockett
for his helpful discussions on the subject.
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p. 8